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,  4 


ELEMENTARY  THEORY 

OF 

EQUATIONS 


/  BY 

LEONARD  EUGENE  DICKSON,  Ph.D. 

CORRESPONDANT   DE   l'iNSTITUT  DE   FRANCE 

PROFESSOR   OF   MATHEMATICS   IN   THE 

UNIVERSITY   OF   CHICAGO 


NEW   YORK 

JOHN  WILEY  &  SONS,  Inc. 

London:    CHAPMAN    &    HALL,    Limited 


Copyright,  1914, 

BY 

LEONARD  EUGENE  DICKSON 


Printed  in  U.  S.  A. 
Stanbope  ipress 

P.    H.CILSON    COMPANY 

BOSTON,  USA.  ^_3j^ 


PREFACE 


The  longer  an  engineer  has  Ijecn  separated  from  his  alma  mater,  the 
fewer  mathematical  formulas  he  uses  and  the  more  he  relies  upon  tables 
and,  when  the  latter  fail,  upon  graphical  methods.  Although  graphical 
methods  have  the  advantage  of  being  ocular,  they  frequently  suffer  from 
the  fact  that  only  what  is  seen  is  sensed.  But  this  defect  is  due  to  the 
kind  of  graphics  used.  With  the  aid  of  the  scientific  art  of  graphing  pre- 
sented in  Chapter  I,  one  may  not  merely  make  better  graphs  in  less  time 
but  actually  draw  correct  negative  conclusions  from  a  graph  so  made, 
and  therefore  sense  more  than  one  sees.  For  instance,  one  may  be  sure 
that  a  given  cubic  equation  has  only  the  one  real  root  seen  in  the  graph, 
if  the  bend  points  lie  on  the  same  side  of  the  a;-axis. 

Emphasis  is  here  placed  upon  Xe^^'ton's  method  of  solving  numerical 
equations,  both  from  the  graphical  and  the  numerical  standpoint.  One 
of  several  advantages  (well  recognized  in  Europe)  of  Newton's  method  over 
Horner's  is  that  it  applies  as  well  to  non-algebraic  as  to  algebraic  equations. 

In  this  elementary  book,  the  author  has  of  course  omitted  the  difficult 
Galois  theory  of  algebraic  equations  (certain  texts  on  which  are  very 
erroneous)  and  has  merely  illustrated  the  subject  of  invariants  by  a  few 
examples. 

It  is  surprising  that  the  theorems  of  Descartes,  Budan,  and  Sturm,  on 
the  real  roots  of  an  equation,  are  often  stated  inaccurately.  Nor  are  the 
texts  in  English  on  this  subject  more  fortunate  on  the  score  of  correct 
proofs;  for  these  reasons,  care  has  been  taken  in  selecting  the  books  to 
which  the  reader  is  referred  in  the  present  text. 

The  material  is  here  so  arranged  that,  before  an  important  general 
theorem  is  stated,  the  reader  has  had  concrete  illustrations  and  often  also 
special  cases.  The  exercises  are  so  placed  that  a  reasonal^ly  elegant  and 
brief  solution  may  be  expected,  without  resort  to  tedious  multiplications 
and  similar  manual  labor,  ^'ery  few  of  the  five  hundred  exercises  are  of 
the  same  nature. 

Complex  numbers  are  introduced  in  a  logical  and  satisfying  manner. 
The  treatment  of  roots  of  unity  is  concrete,  in  contrast  to  the  usual  ab- 
stract method. 

Attention  is  paid  to  scientific  computation,  })oth  as  to  control  of  the 
limit  of  error  and  as  to  securing  maximum  accuracy  with  minimum  labor. 

An  easy  introduction  to  determinants  and  their  application  to  the  solu- 
tion of  systems  of  linear  equations  is  afforded  by  Chapter  XI,  which  is 
independent  of  the  earlier  chapters. 


m61?34 


IV  PREFACE 

Here  and  there  are  given  brief,  ])ut  clear,  outlooks  upon  various  topics 
of  decided  intrinsic  and  historical  interest,  —  thus  putting  real  meat  upon 
the  dry  bones  of  the  subject. 

To  provide  for  a  very  brief  course,  certain  sections,  aggregating  over 
fifty  pages,  are  marked  by  a  dagger  for  omission.  However,  in  compensa- 
tion for  the  somewhat  more  advanced  character  of  these  sections,  they  are 
treated  in  greater  detail. 

In  addition  to  the  large  number  of  illustrative  problems  solved  in  the 
text,  there  are  five  hundred  very  carefully  selected  and  graded  ercercises, 
distributed  into  seventy  sets.  As  only  sixty  of  these  exercises  (falling  into 
seventeen  sets)  are  marked  with  a  dagger,  there  remains  an  ample  number 
of  exercises  for  the  briefer  course. 

The  author  is  greatly  indel^ted  to  his  colleagues  Professors  A.  C.  Lunn 
and  E.  J.  Wilczynski  for  most  valuable  suggestions  made  after  reading 
the  initial  manuscript  of  the  book.  Useful  advice  was  given  by  Professor 
G.  A.  Miller,  who  read  part  of  the  galley  proofs.  A  most  thorough  read- 
ing of  both  the  galley  and  page  proofs  was  very  generously  made  by 
Dr.  A.  J.  Kempner,  whose  scientific  comments  and  very  practical  sugges- 
tions have  led  to  a  marked  improvement  of  the  book.  Moreover,  the 
galleys  were  read  critically  by  Professor  D.  R.  Curtiss,  who  gave  the  author 
the  benefit  not  merely  of  his  A\'ide  knowledge  of  the  subject  but  also  of  his 
keen  critical  ability.  The  author  sends  forth  the  book  thus  emended 
^\ith  less  fear  of  future  critics,  and  with  the  hope  that  it  will  prove  as 
stimulating  and  useful  as  these  five  friends  have  been  generous  of  their 
aid. 

Chicago.  Februury,  1914. 


CONTENTS 


Chap.  Page 

I.   The  Graph  of  an  Equation 1 

II.    Complex  Numbers 18 

III.  Algebraic  and  Trigonometric  Solution  of  Cubic  Equations 31 

IV.  Algebraic  Solution  of  Quartic  Equations 38 

V.   The  Fundamental  Theorem  of  Algebra 47 

VI.   Elementary  Theorems  on  the  Roots  of  an  Equation 55 

VII.   Symmetric  Functions 63 

VIII.   Reciprocal  Equations.     Construction  of  Regular  Polygons.     Trisection  of 

an  Angle 81 

IX.   Isolation  of  the  Real  Roots  of  an  Equation  with  Real  Coefficients 93 

X.   Solution  of  Numerical  Equations 109 

XI.    Determinants.     Systems  of  Linear  Equations 127 

XII.    Resultants  and  Discriminants 150 

Miscellaneous  Exercises 167 

Answers 177 

Index 183 


THEORY    OF    EQUATIONS 


C-1,4) 


CHAPTER  I 

The  Graph  of  an  Equation 

1.  For  purposes  of  review,  certain  terms  already  familiar  to  the  reader 
are  defined  here.  Through  a  point  0,  called  the  origin,  draw  a  horizon- 
tal straight  line  OX  and  a  vertical  straight  line  OY.  These  lines  are 
called  the  axes  of  coordinates;  in  particular,  OX  is  called  the  x-axis. 
Choose  a  convenient  unit  of  length.  Consider  any  point  P  in  the  plane 
and  let  Q  be  the  point  of  intersection  of  the  x-axis  with  the  vertical  line 
through  P.  By  the  abscissa  x  of  the  point  P  is  understood  the  number 
of  units  of  length  in  OQ  in  case  Q  lies  to  the  right  of  the  origin  0;  but,  in 
case  Q  lies  to  the  left  of  0,  x  is  the  negative  of  the  number  of  units  of  length 
in  OQ.  Similarly,  the  ordinate  y  of  the  point  P 
is  the  length  of  PQ  when  P  is  above  the  x-axis, 
but  is  the  negative  of  the  length  of  PQ  when  P 
is  below  the  x-axis.  For  the  point  P  in  Fig.  1, 
x  =+4,  y  =—11.  The  real  numbers  x  and 
y  which  a  point  determines  in  this  manner  are 
called  its  coordinates.  Conversely,  any  pair  of 
real  numbers  determines  a  point. 

Figure  1  shows  the  points  which  represent 
various  pairs  of  values  of  x  and  y,  satisfying 
the  equation 

(1)  ?/  =  x2-6x-3. 

For  example,  the  point  P  represents  the  pair 
of  values  x  =  4,  y  =  —11,  and  is  designated 
(4,   —11).     Since   the  value  of   x  may  be  as- 
signed at  pleasure  and  a  corresponding  value  of  y  is  determined  by 
equation  (1),  there  is  an  infinitude  of  points  representing  pairs  of  values 

1 


(6,-3) 


THEORY  OF  EQUATIONS 


'Cr.  1 


satisfying  the  equation.  These  points  constitute  a  curve  called  the  graph 
of  the  equation. 

In  Fig.  1,  the  curve  intersects  the  x-axis  in  two  points;  the  abscissa 
of  one  point  of  intersection  is  between  6  and  7,  that  of  the  other  point  is 
between  —  1  and  0.  The  x-axis  is  the  graph  of  the  equation  y  =  0.  Thus 
the  abscissas  of  the  intersections  of  the  graph  of  equation  (1)  and  the 
graph  oi  7j  —  0  arc  the  real  roots  of  the  quadratic  equation 
(!')  x2-6x-3  =  0. 

Hence  to  find  graphically  the  real  roots  of  the  last  equation,  we  equate 
the  left  member  to  y  and  use  the  graph  of  the  resulting  equation  (1). 
For  other  methods,  see  §§  16-18. 


EXERCISES 

Find  graphically  the  real  roots  of  .x^  —  6  x  +  7  =  0. 

Discuss  graphically  the  reality  of  the  roots  of  x-  —  Q  x  +  12  =  0. 

Obtain  the  grapli  used  in  Ex.  1  by  shifting  the  graph  in  Fig.  1  ten  units 
upwards,  leaving  the  axes  OX  and  OY  unchanged.  How 
may  we  obtain  similarly  that  used  in  Ex.  2? 

4.   Locate  graphically  the  real  roots  of  x^  +  4  .r-  —  7  =  0- 


2.    Caution  in  Plotting.     If  the  example  set  were 


(2) 


y  =  S  x^  -  14  x^  -  9  .t2  +  11  X 


one  might  use  successive  integral  values  of  x,  obtain 
the  points  (-2,  180),  (-1,  0),  (0,  -2),  (1,  -6), 
(2,  0),  (3,  220),  all  but  the  first  and  last  of  which  are 
shown  (by  crosses)  in  Fig.  2,  and  be  tempted  to  con- 
clude that  the  graph  is  a  U-shaped  curve  approxi- 
mately like  that  in  Fig.  1  and  that  there  are  just  two 
real  roots,  —  1  and  2,  of 

(2')  8x'*  -  Ur*  -  9x2-Mlx -2  =  0. 

But  both  of  th(>se   conclusions  would  be  false.     In 

fact,  the  graj^h  is  a  W-shaped  curve  (Fig.  2)  and  the 

additional  real  roots  arc^  \  and  |. 

This  example  shows  that  it  is  often  necessary  to 

employ  also  values  of  x  which  are  not  integers.  The 
purpose  of  the  exami)le  was,  however,  not  to  point  out  this  o])vious  fact, 
but  rather  to  emphasize  the  chance  of  serious  error  in  sketching  a  curve 


§  3] 


THE  GRAPH  OF  AN  EQUATION 


through  a  number  of  points,  however  numerous.  The  true  curve  between 
two  points  below  the  a:-axis  may  not  cross  the  x-axis,  or  may  have  a 
peak  actually  crossing  the  a;-axis  twice,  or  may  be  an  M-shaped  curve 
crossing  it  four  times,  etc. 


For  example,  the  graph  (Fig.  3)  of 


(3) 


y  =  x^  -\-  4:X"  —  11 


crosses  the  .x-axis  only  once.     But  this  fact  can  not  be  concluded  from 
a  graph  located  by  a  number  of  points,  how- 
ever numerous,  whose  abscissas  are  chosen  at 
random. 

We  shall  find  that  correct  conclusions  re- 
garding the  number  of  real  roots  can  be  de- 
duced from  a  graph  whose  bend  points  (§  3) 
have  been  located. 

We  shall  be  concerned  with  equations  of  the 
form 


aox""  +  aix"-!  + 


+  an-iX  +  a„  =  0 

(ao  ^  0), 


in  which  ao,  Oi,  .  .  .  a„  are  real  constants. 
The  left  member  is  called  a  'polynomial  in  x  of 
degree  n,  or  also  a  rational  integral  function  of  x^ 
and  will  frequently  be  denoted  for  brevity  by 
the  symbol  f(x)  and  less  often  by  /. 


Fig.  3 


3.  Bend  Points.  A  point  (like  M  or  M'  in  Fig.  3)  is  called  a  bend 
point  of  the  graph  of  y  =  f{x)  if  the  tangent  to  the  graph  at  that  point 
is  horizontal  and  if  all  of  the  adjacent  points  of  the  graph  lie  below  the 
tangent  or  all  above  tlie  tangent.  The  first,  but  not  the  second,  condi- 
tion is  satisfied  by  the  point  0  of  the  graph  of  y  =  x^  given  in  Fig.  4 
(see  §  6).  In  the  language  of  the  calculus, /(x)  has  a  (relative)  maximum 
or  minimum  value  at  the  abscissa  of  a  bend  point  on  the  graph  of  y  = 
fix). 

Let  P  =  (x,  y)  and  Q  =  (x  -\-  h,  Y)  be  two  points  on  the  graph, 
sketched  in  Fig.  5,  oi  y  =  f{x).     By  the  slope  of  a  straight  line  is  meant 


THEORY  OF  EQUATIONS 


[Ch.  I 


the  tangent  of  the  angle  between  the  line  and  the  a:-axis  measured  counter- 
clockwise from  the  latter.     In  Fig.  5,  the  slope  of  the  straight  line  PQ  is 

Y-y  _f(x-hh)-fix) 
h  h 


(4) 


Y 

.r 

/ 

Y-y 

h 

0 

X 

Fig.  4  Fig.  5 

For  equation  (3),  fix)  =  x^  +  4  x^  —  11.     Hence 

fix  +  h)  =  ix  +  hy  +  4  (x  +  /i)2  -  11 

=  x'  +  4  x2  -  11  +  (3  a:-  +  8  x)h  +  (3  x  +  4)/r  +  h\ 

The  slope  (4)  of  the  secant  PQ  is  here 

3x-  +  8.'c+  (3.'c  +  4)/i  +  /i2. 

Now  let  the  point  Q  move  along  the  graph  towards  P.  Then  h  approaches 
the  value  zero  and  the  secant  PQ  approaches  the  tangent  at  P.  The 
slope  of  the  tangent  at  P  is  therefore  the  corresponding  limit  3  a--  +  8  a: 
of  the  preceding  expression. 

In  particular,  if  P  is  a  bend  point  the  slope  of  the  tangent  at  P  is  zero 
and  hence  a:  =  0  or  x  =  —  f.  Equation  (3)  gives  the  corresponding 
values  of  y.     The  resulting  points 

il/=  (0,  -11),     iir  =  (-!,  -H) 


§4]  THE  GRAPH  OF  AN  EQUATION  5 

are  easily  shown  to  be  bend  points.  Indeed,  for  re  >  0  and  for  x  between 
-4  and  0,  x"^  {x  +  4)  is  positive,  and  hence /(a;)  >  -11  for  such  values  of 
X,  so  that  the  function  (3)  has  a  relative  minimum  at  a;  =  0.  Similarly, 
there  is  a  relative  maximum  at  a:  =  —  |.  We  may  also  employ  the  general 
method  of  §  8  to  show  that  M  and  M'  are  bend  points.  Since  these  bend 
points  are  both  below  the  a;-axis,  we  are  now  certain  that  the  graph 
crosses  the  a:-axis  only  once. 

The  use  of  the  bend  points  insures  greater  accuracy  to  the  graph  than 
the  use  of  dozens  of  points  whose  abscissas  are  taken  at  random. 

4.   Derivatives.     We  shall  now  find  the  slope  of  the  tangent  to  the 
graph  of  ?/  =  f(x) ,  where  f(x)  is  any  polynomial 
(5)  fix)  =  aox"  +  aia:"-i  +  •  •  •  +  an-ix  +  a„. 

We  need  the  expansion  of  f{x  +  h)  in  powers  of  x.  By  the  binomial 
theorem, 

ao{x  -\-  h)"  =  aox""  +  naoa;"-'/i  +  -^^ aoX"--h^  -\-  .  .  .  ^ 

a,(x  +  h)"-'  =  a^x"-'  +  (n  -  l)a,x^'-%  +  ^"^ ~  ^^j,"'  ~  ^K.x"-^-'  +  •  •  •  , 


a„-2(a:  +  h)-  =  a„-2X-  +  2  an-^xh  +  dn-ih^, 

Qn-liX  +  h)    =   Qn-lX  +  ttn-lh, 

The  sum  of  the  left  members  is  evidently  f(x  +  h).  On  the  right,  the 
sum  of  the  first  terms  (i.e.,  those  free  of  h)  is  f(x).  The  sum  of  the 
coefficients  of  h  is  denoted  by  f'{x),  the  sum  of  the  coefficients  of  §  h-  is 
denoted  hy  f"{x),  •  •  •  ,  the  sum  of  the  coefficients  of 

h/^ 

1-2     ••  A; 
is  denoted  by  /(^'^  (x) .     Thus 

(6)  fix)    =  naox"-'  +  (n  -  l)aiX--'~  +  •  •  •  +  2  a„_2a:  +  a„_i, 

(7)  fix)  =  nin  -  1)  ao.x"-2  +  in  -  l)(n  -  2)ayX-'  +  •  •  •  +  2a„_2, 
etc.  Hence  we  have 

(8)  f(x  +  A)  =  fix)  +f'(x)  k  +/"Wj^ 

+/"'Wr:|:3+---+/'"'Wi-T2^- 


6  THEORY  OF  EQUATIONS  [Ch.  I 

This  formula  (8)  is  known  as  Taylor's  theorem  for  the  present  case  of 
a  polynomial  f{x)  of  degree  n.  We  call  f'{x)  the  (first)  derivative  of 
f{x),  and  fix)  the  second  derivative  of  f{x),  etc.  Concerning  the 
fact  that  f"{x)  is  the  first  derivative  of  f'{x)  and  that,  in  general,  the 
A:th  derivative /(*■")  (a^)  of  f{x)  equals  the  first  derivative  of  /^'■'"'K^);  see 
Exs.  6-9  of  the  next  set. 

In  view  of  (8),  the  limit  of  (4)  as  h  approaches  zero  isf'{x).  Hence 
f'{x)  is  the  slope  of  the  tangent  to  the  graph  of  y  =  f{x)  at  the  point  {x,  y). 

In  (5)  and  (6),  let  every  a  be  zero  except  a^.  Thus  the  derivative  of 
aox"  is  naoX"~\  and  hence  is  obtained  by  multiplying  the  given  term  by 
its  exponent  n  and  then  diminishing  its  exponent  by  unity.  For  example, 
the  derivative  of  2  x^  is  6  x~. 

Moreover,  the  derivative  of  /(.r)  equals  the  sum  of  the  derivatives  of 
its  separate  terms.  Thus  the  derivative  of  a;^  +  4  x^  —  1 1  is  3  x-  +  8  a:, 
as  found  also  in  §  3. 

5.  Computation  of  Polynomials.  The  lalwr  of  computing  the  value 
of  a  polynomial  J{x)  for  a  given  value  of  x  may  be  much  shortened  by 
a  simple  device.     To  find  the  value  of 

a:^  +  3  ^2  -  2  .r  -  5 

for  X  =  2,  we  note  that  x^  =  x-x-  =  2  x^,  so  that  the  sum  of  the  first  two 
terms  is  5  x"^.     This  latter  equals  5  •  2  x  or  10  a;,  adding  this  to  the  next 
term  —2  x,  we  get  8  a;  or  16.     The  final  result  is  therefore  11. 
Write  the  coefficients  in  a  line.     Then  the  work  is: 

1  3-2        -  5  ^ 

2         10  16 


1  5  8  11. 


In  case  not  all  the  intermediate  powers  of  x  occur  among  the  terms  of 

f{x),  the  missing  powers  are  considered  as  having  the  coefficients  zero. 

Thus  the  value  —61  of  2  a;^  —  x^  +  2  x  —  1   for  a:  =  —2  is  found  as 

follows: 

2  0-1  0  2-1  1-2 

-4  8-14  28      -60 

2-4  7      -14  30       -61. 

For  another  manner  of  presenting  this  method  see  Ch.  X,  §  4. 


THE  GRAPH  OF  AN  EQUATION 


EXERCISES 


1.  The  slope  of  the  tangent  to  ?/  =  8  x^  —  22  a:^  +  13  a;  —  2  at  {x,  y)  is 
24x2  _  44a;  +  13.  The  bend  pomts  are  (0.37,0.203),  (1.46,  -5.03),  approxi- 
mately.    Draw  the  graph. 

2.  The  bend  points  of  y  =  r'  -  2  x  -  b  are  (.82,  -6.09),  (-.82,  -3.91), 
approximately.     Draw  the  graph  and  locate  the  real  roots. 

3.  Find  the  bend  points  of  ?/  =  x^  +  6  x^  +  8  .r  +  8.     Locate  the  real  roots. 

4.  Locate  the  real  roots  of  /(.c)  =  x*  +  x^  —  x  —  2  =  0.  The  abscissas  of 
the  bend  points  are  the  roots  of  f  {x)  =  4  .r^  +  3  x^  —  1  =  0.  The  bend  points 
of  y  =  /'(x)  are  (0,  —1)  and  (  — ^,  —  f),  so  that  /'(x)  =  0  has  a  single  real  root 
(it  is  just  less  than  ^).  The  single  bend  point  of  y  =  /(x)  is  (^,  —  f  J),  approxi- 
mately. 

5.  Locate  the  real  roots  of  x^  —  7  x^  —  3  x-  +  7  =  0. 

6.  /"(x),  given  by  (7),  is  the  first  derivative  of /'(x). 

7.  If  /(x)  =  /i(x)  +/2(x),  the  ^th  derivative  of  /  equals  the  sum  of  the  A:th 
derivatives  of  /i  and  f^.     Use  (8) . 

8.  f^'^^i^x)  equals  the  first  derivative  of  f'^^~^\x).  Hint:  prove  this  for  /  =ax"'; 
then  prove  that  it  is  true  for  /  =  /i  +  /e  if  true  for  /i  and  /2. 

9.  Find  the  third  derivative  of  x''  +  5  x^  ]:)y  forming  successive  first  derivatives; 
also  that  of  2  x^  —  7  x'  +  x. 

10.  The  derivative  of  gk  is  g'k  +  gk'.  Hint :  multiply  the  members  of  g{x  +  A )  = 
gi^)  +  f7'(-c)  h  +   •  •  •  and   k{x  +  h)  =  ^(x)  +  k'{x)  h  +   •  •  •  and    use    (8)    for 

f=gk. 

6.  Horizontal  Tangents.  If  (x,  y)  is  a  bend  point  of  the  graph  of 
y  =  fi^)>  then,  by  definition,  the  slope  of  the  tangent  at  (x,  y)  is  zero. 
Hence  (§  4),  the  abscissa  a;  is  a  root  of  f'(x)  =  0.  In  Exs.  1-5  of  the 
preceding  set,  it  was  true  that,  conversely,  any  real  root  of  f'(x)  =  0 
is  the  abscissa  of  a  bend  point.  However,  this  is  not  always  the  case. 
We  shall  now  consider  in  detail  an  example  illustrating  this  fact.  The 
example  is  the  one  merely  mentioned  in  §  3  to  indicate  the  need  of  the 
second  requirement  made  in  our  definition  of  a  bend  point. 

The  graph  (Fig.  4)  of  ?/  =  x^  has  no  bend  point  since  x^  increases  when 
X  increases.  Nevertheless,  the  derivative  3  x-  of  x^  is  zero  for  the  real 
value  a;  =  0.  The  tangent  to  the  curve  at  (0,  0)  is  the  horizontal  line 
?/  =  0.  It  may  be  thought  of  as  the  limiting  position  of  a  secant  through 
0  which  meets  the  curve  in  two  further  points,  seen  to  be  equidistant 
from  0.  When  one,  and  hence  also  the  other,  of  the  latter  points  ap- 
proaches 0,  the  secant  approaches  the  position  of  tangency.  In  this 
sense  the  tangent  at  0  is  said  to  meet  the  curve  in  three  coincident 
points,  their  abscissas  being  the  three  coinciding  roots  of  x^  =  0.     In  the 


8  THEORY  OF  EQUATIONS  [Ch.  I 

usual  technical  language  which  we  shall  employ  henceforth,  x^  =  0  has 
the  triple  root  x  =  0.  The  subject  of  bend  points,  to  which  we  recur  in 
§  8,  has  thus  led  us  to  a  digression  on  the  important  subject  of  double 
roots,  triple  roots,  etc. 

7.   Multiple  Roots.     In  (8)  replace  a:  by  a  and  h  by  x  —  a.     Then 

(9)       /(a:)=/(a)+/'(a)(a;-a)+r(a)-^^f^+r'(a)^'^~"^'   ' 


1-2       '  •'     '  M  •  2  ■  3    ' 

Thus  the  constant  remainder  obtained  by  dividing  any  polynomial  f{x) 
by  a;  —  a  is  /(a),  a  fact  known  as  the  Remainder  Theorem.  In  par- 
ticular, if  /(«)  =  0,  f{x)  has  the  factor  x  —  a.  This  proves  the  Factor 
Theorem:  If  a  is  a  root  of /(x)  =  0,  then  x  —a  is  a  factor  of /(a:). 

The  converse  is  true:  If  x  —  a  is  a  factor  of /(.t),  then  a  is  a  root  of 
f{x)  =  0.  In  case  f{x)  has  the  factor  {x  —  a)-,  but  not  the  factor 
{x  —  aY,  a  is  called  a  double  root  of /(x)  =  0.  In  general,  if  f{x)  has 
the  factor  {x  —  a)'",  but  not  the  factor  {x  —  a)"'+\  a.  is  called  a  multiple 
root  of  multiplicitij  m  oi  f{x)  —  0,  or  an  ?»-fold  root.  Thus,  4  is  a  simple 
root,  3  a  double  root  and  —2a  triple  root  of 

7{x-4:)ix-3y-(x-\-2y  =  0. 

This  algebraic  definition  of  a  multiple  root  is  in  fact  equivalent  to  the 
geometrical  definition,  given  for  a  special  case,  in  §  6. 

The  second  member  of  (9)  is  divisible  by  (x  —  a)-  if  and  only  if /(a)  =  0, 
/'(a)  =  0,  and  is  divisible  by  (x  —  a)^  if  and  only  if  also  /"(a)  =  0,  etc. 
Hence  a  is  a  double  root  of  f(x)  =  0  if  and  only  if  f(a)  =  0,  f'(a)  =  0, 
/"  (a)  ?^  0;   a  is  a  root  of  multiplicity  ni  if  and  only  if 

(10)         f{a)  =  0,  f'{a)  =  0,/"(a)  =  0,   •  •   •   ,  /("-')(«)  =  0,  /("')(«)  ^  Q. 

For  example,  zero  is  a  triple  root  of  x*  +  2  x'  =  0  since  the  first  and  second 
derivatives  are  zero  for  x  =  0,  while  the  third  derivative  24  x  +  12  is  not. 

If /(x)  and/'(x)  have  the  common  factor  (x  —  a)'"~^,  but  not  (x  —  a)'", 
where  m  =  2,  then  a  is  a  root  of  f{x)  =  0  of  multiplicity  7n.  For,  a  is 
a  root  of  multiplicity  at  least  ni  —  1  of  both  /(.r)  =  0  and  /'(x)  =  0,  so 
that  the  equalities  in  (10)  hold;  also  /("■)(«)  ^  0  holds,  since  otherwise  a 
would  be  a  root  of  l)oth  /(x)  =  0  and  /'(x)  =  0  of  multiplicity  m  or  greater, 
and  (x  —  a)""  would  be  a  common  factor.  Hence  if  f{x)  andf'{x)  have  a 
greatest  common  divisor  g{x)  involving  x,  a  root  of  g(x)  =0  of  multiplicity 


§  8]  THE  GRAPH  OF  AN  EQUATION  9 

m  —  \  is  a  root  of  f(x)  =  0  of  multiplicity  m,  and  conversely  any  root  of 
f{x)  =  0  of  niulti-plicity  m  is  a  root  of  g(x)  =  0  of  multiplicity  m  —  1.  The 
last  fact  follows  from  relations  (10),  which  imply  that  a  is  a  root  of 
f'{x)  =  0  of  multiplicity  m  —  1,  and  hence  that  f{x)  and  f'{x)  have  the 
common  factor  {x  —  a)"'"^,  but  not  (x  —  a)'". 

In  view  of  this  theorem,  the  problem  of  finding  all  the  multiple  roots 
oi  f{x)  =  0  and  the  multiplicity  of  each  multiple  root  is  reduced  to  the 
problem  of  finding  the  roots  of  g{x)  =  0  and  the  multiplicity  of  each. 

For  example,  let  J{x)  =  x^  —  2  x^  —  4  x  +  8.    Then 

/'(x)  -  3  x2  -  4 X  -  4,     9/(x)  =  /'(x)  (3  x  -  2)  -  32  (x  -  2). 

Since  x  —  2  is  a  factor  oi  fix)  it  may  be  taken  to  be  the  greatest  common  divisor 
of /(x)  and/'(x),  as  the  choice  of  the  constant  factor  c  in  c(x  —  2)  is  here  immaterial. 
Hence  2  is  a  double  root  of  /(x)  =  0,  while  the  remaining  root  —2  is  a  simple  root. 

EXERCISES 

1.  x^  —  7  x^  -f  15  X  —  9  =  0  has  a  double  root. 

2.  X*  —  8x^+16  =  0  has  two  double  roots. 

3.  X*  —  6x2  —  8x  —  3  =  0  ]^jjg  a  triple  root. 

4.  Test  x^  -  8  x»  +  22  x2  -  24  X  +  9  =  0  for  multiple  roots. 

5.  Test  x^  —  6  .r^  +  11 X  —  6  =  0  for  multiple  roots. 

8.  Inflexion  and  Bend  Points.  The  equation  of  the  tangent  to  the 
graph  oi  y  =  f{x)  at  the  point  {a,  jS)  on  it  is 

y=na){x-a)-V^  [^  = /(«)]• 

For  the  abscissas  of  its  intersections  with  the  graph  oi  y  =  f(x),  we  have, 
from  (9), 

r(a)^^j^  +  r'(a)^^;^'+-  •  •  =0. 

If  a:  is  a  root  of  multipHcity  m  of  this  equation,  the  point  (a,  /3)  is  counted 
as  m  coincident  points  of  intersection  of  the  tangent  and  the  curve  (just 
as  in  the  example  in  §  6).     This  will  be  the  case  if  and  only  if  * 

(11)         f"ia)  =0,  f"'{a)  =  0,  .   .   .   ,  /(-i)(«)  =  0,  /(-)(«)  ^  0. 

For  example,  if  /(x)  =  x*  and  a  =  0,  then  m  =  4.  The  graph  oi  y  =  x*  is  a. 
U-shaped  curve,  whose  intersection  with  the  tangent  (x-axis)  at  (0,  0)  is  counted 
as  four  coincident  points  of  intersection. 

*  If  m  =  2,  only  the  last  relation  of  the  set  is  retained:  /"(a)  5^  0. 


10  THEORY  OF  EQUATIONS  [Ch.  I 

If  m  is  even,  the  points  of  the  curve  in  the  vicinity  of  the  point  of 
tangency  {a,  /3)  are  all  on  the  same  side  of  the  tangent  and  the  point  (a,  /3) 
is,  by  the  definition  in  §  3,  a  bend  point.  But  if  ni  is  odd  (wi  >  1),  the 
curve  crosses  the  tangent  at  the  point  of  tangency  {a,  jS)  and  this  point 
is  called  an  inflexion  point,  and  the  tangent  an  inflexion  tangent.  To 
simplify  the  proof,  take  {a,  /3)  as  the  new  origin  of  coordinates  and  the 
tangent  as  the  new  a;-axis.     Then  the  new  equation  of  the  curve  is 

y  =  cx"^  +  dx'^+i  +  •  •  •  {c^0,m  =  2). 

For  X  sufficiently  small  numerically,  y  has  the  same  sign  as  ex""  (§  11). 
Thus  if  m  is  even,  the  points  of  the  curve  in  the  vicinity  of  the  origin  are 
all  on  the  same  side  of  the  a:-axis.  But  if  m  is  odd,  the  points  with  small 
positive  abscissas  lie  on  one  side  of  the  a:-axis  "and  those  with  numerically 
small  negative  abscissas  lie  on  the  opposite  side. 

For  example,  (0,  0)  is  a  bend  point  of  the  graph  of  y  =  x*.  But  (0,  0)  is  an 
inflexion  point  of  the  graph  (Fig.  4)  of  y  =  .T^  and  the  inflexion  tangent  y  =  0 
crosses  the  curve  at  (0,  0).  Here/"(0)  =  0,  /'"(O)  =  6,  so  that  ?«  =  3,  in  accord 
with  the  evident  fact  that  a;^  =  0  has  the  root  zero  of  multipUcity  3. 

We  have,' therefore,  in  the  evenness  or  oddness  of  m  in  (11)  a  practical 
test  to  decide  which  roots  a  of  f'{x)  =  0  are  abscissas  of  bend  points 
and  which  are  abscissas  of  inflexion  points  with  horizontal  inflexion 
tangents. 

EXERCISES 

1.  If  fix)  =  3  x^  +  5  x3  +  4,  the  only  real  root  of  fix)  =  0  is  x  =  0.  Show 
that  (0,  4)  is  an  inflexion  point,  and  thus  that  there  is  no  bend  point  and  hence 
that  fix)  =  0  has  a  single  real  root. 

2.  x^  —  3  x^  ■]-  3x  +  c  =  0  has  an  inflexion  point,  but  no  bend  point. 

3.  x^  —  10  x'  —  20  x^  —  15  .r  +  c  =  0  has  two  bend  points  and  no  horizontal 
inflexion  tangents. 

4.  3  x^  —  40  x^  +  240  x  +  r  =  0  has  no  bend  point,  but  has  two  horizontal 
inflexion  tangents. 

5.  Any  function  x^  —  3  ox^  +  •  •  -of  the  third  degree  can  be  written  in  the 
fonn/(x)  =  (x  —  a)^  +  ax  +  6.  The  straight  Hne  having  the  equation  y  =  ax-\-b 
meets  the  graph  oi  y  =  fix)  in  three  coincident  points  with  the  abscissa  a  and 
hence  is  an  inflexion  tangent.  If  we  take  new  axes  of  coordinates  parallel  to  the 
old  and  intersecting  at  the  new  origin  («,  0),  i.e.,  if  we  make  the  transformation 
x  =  X-\-a,  y=Y,  of  coordinates,  we  see  that  the  equation  /(x)  =  0  becomes  a 
reduced  cubic  equation  X^  +  pX  +  q  =  0  (cf.  Ch.  III). 

6.  Find  the  inflexion  tangent  to  y  =  x^  +  6  .r-  —  3  x  +  1  and  transform 
x^  +  Gx^  —  3x  +  l  =0  into  a  reduced  cubic  equation. 


§  101  THE  GRAPH  OF  AN  EQUATION  11 

9.   Real  Roots  of  a  Cubic  Equation.     It  suffices  to  consider 

J{x)  =  x'  -  3lx  +  q  (1^0), 

in  view  of  Ex.  5  above.     Then  /'=  3  {x'~  -  I),  f"=  Qx.     in<  0,  there 
is  no  bend  point  and  the  cubic  equation  f(x)  =  0  has  a  single  real  root. 
U  I  >  0,  there  are  two  bend  points 

(Vl,  q-2l  VI),  {-Vl,q-^2lVl) 

and  the  graph  oi  y  =  f{x)  is  evidently  of  one  of  the  three  types: 


If  the  equality  sign  holds  in  the  first  or  second  case,  one  of  the  bend 
points  is  on  the  a:-axis  and  the  cubic 
equation  has  a  double  root;  the  condi- 
tion is  that  q2  _  4  p  =  o.  The  third 
case  is  fully  specified  by  the  condition 
g2  <  4  P,  which  implies  that  Z  >  0. 
Hence  x^  —  3  Ix  -\-  q  =  0  has  three  dis- 
tinct real  roots  if  and  only  if  q-  <  4:  P, 
a  single  real  root  if  and  only  if  q-  >  4  Z^; 
and  a  double  root  {necessarily  real)  if  and  only  if  q^  =  4:  P. 


Fig.  8 


EXERCISES. 
Apply  the  criterion  to  find  the  number  of  real  roots  of: 

1.  a:3  +  2a; -4  =  0.  2.   a;^  -  7x  +  7  =  0.  3.z^-2x-l  =  0. 

4.   x^  -  3  X  +  2  =  0.  5.   x^  +  6  x2  -  3  X  +  1  =  0. 

6.   The  inflexion  point  of  ?/  =  x^  —  3  /x  +  g  is  (0,  q). 


lO.f   Trinomial  Equations. 

For  m  and  n  positive  odd  integers,  m  >  n,  let 

fix)  =  X"'  +  px"  +  q 


(p  ^  0). 


12  THEORY  OF  EQUATIONS  tCn.  I 

Here  x  =  0  is  a  root  of /'(x)  =  0  only  when  n  >  I  and  then  tlio  tangent  at  (0,  q) 
is  the  horizontal  inflexion  tangent  ?/  =  </,  as  shown  by  (11)  with  ni  replaced  Ijy  n, 
or  directly  from  the  fact  that  zero  is  a  root  of  odd  multiplicity  n  of  x'"  +  /jx"  =  0. 
Hence  in  no  case  is  zero  the  abscissa  of  a  bend  point. 

If  p  >  0,  /'  has  no  real  root  except  x  =  0.  Thus  there  is  no  bend  point  and 
hence  a  single  real  root  of  /(x)  =  0. 

If  p  <  0,  there  are  just  two  bend  points,  their  abscissas  being  b  and  —b,  where 
b  is  the  single  positive  real  root  of  6™~"  =  —np/m.  The  bend  points  are  on  the 
same  side  or  opposite  sides  of  the  x-axis  according  as 

m  =  q  +  pb"(l  -  ~^,  fi-b)  =  q  -  pb"  (l  -  ^) 

are  of  like  signs  or  opposite  signs.  The  number  of  real  roots  is  1  or  3  in  the  respec- 
tive cases.  Hence  there  are  three  distinct  real  roots  if  and  only  if  the  positive 
number 


exceeds  both  q  and  —q,  i.e.,  if 


-p5"{l-^ 


-p-b"  > ^• 

m  m  —  n 


The  first  member  equals  ft",  so  that  its  (m  —  ?i)th  power  is  the  ?«th  power  of 
^m-n  —  —np/m.     Hence  the  conditions  are  equivalent  to 


-(f 


+1  -'"'- 

)n  —  n 


EXERCISES  t 
l.t   x^  +  px  +  7  =  0  has  three  distinct  real  roots  if  and  only  if 


(Sf 


o>(:;)+.. 


2.t   If  p  and  q  are  positive,  x-"'  —  p.v-"  +  q  =  0  has  four  distinct  real  roots, 
two  pairs  of  equal  roots,  or  no  real  root,  according  as 


f^y"_f_i^r">0,=0,cr<0. 

\  m  I        \m  —  n/ 


11.  Continuity  of  a  Polynomial.  Hitherto  we  have  located  certain 
points  of  the  graph  of  y  =  /(x),  where  fix)  is  a  polynomial  in  x  wdth  real 
coefficients,  and  taken  the  liberty  to  join  them  by  a  continuous  curve. 


§  12]  THE  GRAPH  OF  AN  EQUATION  13 

The  polynomial  /(x)  in  the  real  variable  x  shall  be  called  continuous  at 
X  =  a,  where  a  is  a  real  constant,  if  the  difference 

D=fia  +  h)-  /(a) 

is  numerically  less  than  any  assigned  positive  number  p  for  all  real  values 
of  h  sufficiently  small  numerically. 

We  shall  prove  that  any  polynomial  f{x)  with  real  coefficients  is  con- 
tinuous at  X  =  a,  where  a  is  any  real  constant. 

The  proof  rests  upon  Taylor's  formula  (8),  which  gives 

Z)./'(a)„+0|),.+  ...+_|^,,. 

Denote  by  g  the  greatest  numerical  value  of  the  coefficients  of  h, 
h?,  .  .  .  ,  h".  For  h  numerically  less  than  k,  where  k  <  1,  we  see  that  D 
is  numerically  less  than 

g{k-\-k''-\-   -  ■  ■   +k-)<g--^<p,     Uk<      ^     - 


1  -  fc      '^'  P  +  g 

The  same  proof  shows  that,  if  ai,  .  .  .  ,  a„  are  real,  aih  +  •  •  •  +  cin^'* 
is  numerically  less  than  an  assigned  positive  number  p  for  all  real  values 
of  h  sufficiently  small  numerically. 

.12.  Theorem.  //  the  coefficients  of  the  polynomial  f{x)  are  real  and  if 
a  and  b  are  real  numbers  such  that  /(a)  and  f{b)  have  opposite  signs,  the 
equation  f{x)  =  0  has  at  least  one  real  root  between  a  and  b;  in  fact,  an  odd 
number  of  such  roots,  if  an  m-fold  root  is  counted  as  m  roots. 

The  only  argument*  given  here  is  one  based  upon  geometrical  intui- 
tion.    We  are  stating  that,  if  the  points 

(a, /(a)),     (6,/(6)) 

lie  on  opposite  sides  of  the  x-axis,  the  graph  oi  y  =  f(x)  crosses  the 
X-axis  once,  or  an  odd  number  of  times,  between  the  vertical  lines  through 
these  two  points.  Indeed,  the  part  of  the  graph  between  these  verticals 
is  a  continuous  curve  having  one  and  only  one  point  on  each  intermediate 
vertical  line,  since  the  function  has  a  single  value  for  each  value  of  x. 
This  would  not  follow  for  the  graph  of  y^  =  x. 

*  An  arithmetical  proof  based  upon  a  refined  theory  of  irrational  numbers  is  given 
in  Weber's  Lehrbuch  der  Algebra,  ed.  2,  vol.  1,  p.  123. 


14  THEORY  OF  EQUATIONS  (Ch.  1 

13.  Sign  of  a  Polynomial.     Given  a  polynomial 

f{x)  =  ooa:"  +  GiX"-^  +  •  •  •  +  On  (oo  5^  0) 

with  real  coefficients,  we  can  find  a  positive  number  P  such  that  f{x)  has 
the  same  sign  as  aoX"  when  x  >  P.     In  fact, 

f{x)=x^^{ao-h<t>),     </>  =  ^  +  |+  •••  +^- 

By  the  last  result  in  §  11,  the  numerical  value  of  ^  is  less  than  that  of  Qq 
when  l/x  is  positive  and  less  than  a  sufficiently  small  positive  number, 
say  l/P,  and  hence  when  x  >  P.  Then  Oo  +  0  has  the  same  sign  as  Qo, 
and  hence  f(x)  the  same  sign  as  Oox". 

The  last  result  holds  also  when  a:  is  a  negative  number  sufficiently  large 
numerically.  For,  if  we  set  x  =  —X,  the  former  case  shows  that /(  —  A") 
has  the  same  sign  as  (  — l)"aoA'"  when  A'  is  a  sufficiently  large  positive 
number. 

We  shall  therefore  say  briefly  that,  for  x  =  +00,  f(x)  has  the  same 
sign  as  oo;  while,  for  x  =  —  00 ,  f{x)  has  the  same  sign  as  ao  if  n  is  even, 
but  the  sign  opposite  to  ao  if  n  is  odd. 

EXERCISES 

1.  x^  +  ax^  -\-  bx  —  4:  =^  0  has  a  positive  real  root  [use  .r  =  0  and  x  =  +^]- 

2.  x^  +  ax^  +  6x  +  4  =  0  has  a  negative  real  root  [use  x  =  0  and  x  =  —  xj. 

3.  If  oo  >  0  and  n  is  odd,  oox"  +•••+««  =  0  has  a  real  root  of  sign  opposite 
to  the  sign  of  a„  [use  x=  —00,  0,  +00]. 

4.  X*  +  ttx^  -\-  bx^  -\-  ex  —  4:  =  0  has  a  positive  and  a  negative  root. 

5.  Any  equation  of  even  degree  n  in  which  the  coefficient  of  x"  and  the  con- 
stant term  are  of  opposite  signs  has  a  positive  and  a  negative  root. 

14.  The  accuracy  of  a  graph  of  y  =  f{x)  can  often  be  tested  and 
important  (;on(;lusions  drawn  from  it  by  use  of  the 

Theorem.  No  straight  line  crosses  the  graph  of  y  =  J{x)  in  more  than 
n  points  if  the  degree  n  of  the  polynomial  f{x)  exceeds  unity. 

A  vertical  line  x  =  c  crosses  it  at  the  single  point  (c, /(c)).  A  non- 
vertical  line  is  the  grai)h  of  an  equation  y  =  mx  +  h  of  the  first  degree, 
and  the  abscissas  of  the  points  of  crossing  are  the  roots  of  mx  +  6  =  f{x). 
The  proof  may  now  be  completed  by  using  the  next  theorem. 


§  15,  16]  THE  GRAPH  OF  AN  EQUATION  15 

15.  Theorem.     An  equation  of  degree  n, 

f(x)  =  aox"  +  aix"-i  +  •  •  •  +  a„  =  0'  (oq  ^  0), 

cannot  have  more  than  n  distinct  roots. 

Suppose  that  it  has  the  distinct  roots  ai,  .  .  .  ,  «„,  a.  By  the  Factor 
Theorem  (§  7),  x  —  ai  is  a  factor  of /(x),  so  that 

f{x)  =  {x-  ai)  Q(x), 

where  Q(x)  is  a  polynomial  of  degree  n  —  1.     Let  x  =  0:2.     We  see  that 

^("2)  —  0,  so  that  as  before 

Q(x)  =  {x  -  02)  Qi(x),    f{x)  =  (x  -  ai){x  -  ai)  Qi(x). 
Proceeding  in  this  manner,  we  get 

f{x)  ^  ao{x  -  ai)(x  -  0:2)   ...   (x  -  a„). 

For  the  root  a,  the  left  member  is  zero  and  the  right  is  not  zero.  Hence 
our  supposition  is  false  and  the  theorem  true. 

EXERCISES 

1.  The  curve  in  Fig.  3,  representing  a  cubic  function,  does  not  cross  the  x-axis 
at  a  second  point  further  to  the  right,  nor  does  the  part  starting  from  M'  and 
running  downwards  to  the  left  later  ascend  and  cross  the  x-axis. 

2.  The  curve  in  Fig.  2,  representing  a  quartic  function,  has  only  the  four  cross- 
ings shown. 

3.  Form  the  cubic  equation  having  the  roots  0,  1,2. 

4.  Form  the  quartic  equation  having  the  roots  ±1,  ±2. 

5.  If  Qqx"  +  •  •  •  =  0  has  more  than  n  distinct  roots,  each  coefficient  is  zero. 
When  would  the  theorem  in  §  14  fail  if  n  =  1? 

6.  If  two  polynomials  in  x  of  degree  n  are  equal  for  more  than  n  distinct  values 
of  X,  they  are  identical. 

7.  An  equation  of  degree  n  cannot  have  more  than  n  roots,  a  root  of  multiplicity 
m  being  counted  as  rn  roots. 

16.  Graphical  Solution  of  a  Quadratic  Equation.     If 

(12)  x2  -  ax  +  6  =  0 

has  real  coefficients  and  real  roots,  the  roots  may  be  constructed  by  the 
use  of  ruler  and  compasses,  i.e.,  by  elementary  geometry. 


16 


THEORY  OF  EQUATIONS 


[Ch.  I 


Draw  a  circle  having  as  a  diameter  the  Une  BQ  joining  the  points 
B  =  (0,  l)and  Q  =  (a,  6);   the  abscissas  ON  and  OM  of  the  points  of 

intersection  of  this  circle  with  the  x-axis  are 
the  roots  of  (12). 

The  center  of  the  circle  is  (a/2,  (6  +  l)/2). 
The  square  of  BQ  is  a^  +  (b  —  iy.  Hence  the 
equation  of  the  circle  is 


(^-5 


%/,_^  +  i- 


<t)-M 


2    /       V2/    '  V    2 

Setting  y  =  0,  we  get  (12). 
Fig.  9  If  we  do  not  insist  upon  a  solution  by 

ruler  and  compasses,  we  may  plot  the  par- 
abola y  =  x^  and  draw  the  straight  line  y  =  ax  —  h;  if  these  intersect, 
the  abscissas  of  the  points  of   intersection   are  the  real  roots  of  (12). 

17.   The  method  last  used  enables  us  to  solve  graphically 

x'  —  ax  +  6  =  0. 

We  have  merely  to  employ  the  abscissas  of  the  intersections  of  the  graph 
(Fig.  4)  of  2/  =  x^  with  y  =  ax  —  b.     For  the  quartic  equation 

z'  +  Az''-\-Bz  +  C  =  0,A>0, 
set  0  =  X  VI;  we  get        x*  +  x^  -  ax  +  6  =  0. 

We  now  employ  the  graphs  of  y  =  x*  +  x^,  y  =  ax  —  h. 

EXERCISES 


Solve  by  each  of  the  two  methods 

1.   x2-5x  +  4  =  0.  2.   x2  +  5a;  +  4  =  0. 

4.   x2-5x-4  =  0.  5.   x2-4x  +  4  =  0. 

Solve  graphically  the  cubic  equations 


x2  +  5  x  -  4  =  0. 
x2  -  3  X  +  4  =  0. 


7.  x3  -  3  X  +  1  =  0. 


x3  +  2  X  -  4  =  0. 


9.   x'  -  7  X  +  7  =  0. 


10.  Find  graphically  the  cube  roots  of  20,  -20,  200. 

11.  State  in  the  language  of  elementary  geometry  the  construction  of  Fig.  9 
and  prove  that  OC  =  TQ  =  b,  TD  =  OB  =  1,  chord  BN  =  chord  DM,  ON  =  MT, 
ON  +  OM  =  a,  ON  •  OM  =  OC'OB  =  b.    Why  are  OM  and  ON  the  roots  of  (12)? 

12.  Any  reduced  cubic  equation  x^  =  px  -\-  q  can  be  solved  by  use  of  a  fixed 
parabola  x^  =  y  and  the  circle  x^  +  y^  =  (/x  +  (p  +  l)y.     (Descartes.) 

13.  x*  =  px-  +  qx  +  r  can  be  solved  by  use  of  a  fixed  parabola  x^  =  y  and  the 
circle  x-  -\-  y"^  —  qx  -\-  {p  -\-  \)y  -\-  r.     (Descartes.) 

14.  Solve  the  cubics  in  Exs.  7-9  by  the  method  of  Ex.  12. 

15.  Solve  X*  =  25  x^  -  60  X  +  36  by  the  method  of  Ex.  13. 


i  181  THE  GRAPH  OF  AN  EQUATION  17 

18. t   Tlie  approximate  values  of  the  real  roots  of  a  cubic  equation 

2^  +  p2  +  g  =  0 

may  be  found  by  a  graphical  method  due  to  C.  Runge.*  We  assign 
equidistant  values  to  z.  For  each  z,  we  have  a  linear  equation  in  p  and  q 
which  therefore  represents  a  straight  line  when  p  and  q  are  taken  as  rec- 
tangular coordinates.  On  a  diagram  showing  these  lines  we  may  locate 
approximately  the  line  (and  hence  the  values  of  z)  corresponding  to 
assigned  values  of  p  and  q.  The  method  applies  also  to  any  equation 
involving  two  parameters  linearly. 

For  the  solution  of  a  numerical  cubic  equation  by  means  of  the  slide 
rule  (and  an  account  of  the  use  of  the  latter),  see  pp.  43-48  of  the  book 
just  cited. 

*  Graphical  Methods,  Columbia  University  Press,  1912,  p.  59  (also,  Praxis  der 
Gleichungen,  Leipzig,  1900,  p.  156).  Earlier  by  L.  Lalanne,  Comptes  Rendus  Acad.  Sc. 
Paris,  81,  1875,  p.  1186,  p.  124.3;  82,  1876,  p.  1487;  87,  1878,  p.  157,  and  in  Notices 
r^unies  par  le  Ministere  des  travaux  .  .  .  exposition  univ.  Paris,  1878. 


CHAPTER  II 

Complex  Numbers 


(For  a  briefer  course,  this  chapter  may  be  begun  with  §  5.) 

1. 1  Vectors  from  a  Fixed  Origin  0.  A  directed  segment  of  a  straight 
fine  is  called  a  vector.  We  shall  employ  only  vectors  from  a  fixed  initial 
point  0. 

The  sum  of  two  vectors  OA  and  OC  is  defined  to  be  the  vector  OS, 

where  S  is  the  fourth  vertex  of  the  par- 
allelogram having  the  fines  OA  and  OC 
as  two  sides.  In  case  A  coincides  with  0, 
the  vector  OA  is  said  to  be  zero;  then 
OS  =  OC. 

A  force  of  given  magnitude  and  given  dir- 
ection is  conveniently  represented  by  a  vector. 
By  a  fundamental  principle  of  mechanics,  two 
forces,  represented  by  the  vectors  OA  and  OC, 
have  as  their  resultant  a  force  represented  by 
the  vector  OS,  as  in  Fig.  10.  Thus  if  two  forces 
their  resultant  is  represented  by  the  sum  of 


Fig.  10 

are  represented  by  two  vectors, 
the  vectors. 


When  referred  to  rectangular  axes  OA"  and  OY,  let  the  point  A  have  the 
coordinates  OE  =  a,  EA  =  h,  and  the  point  C  the  coordinates  OF  =  c, 
FC  =  d.  Draw  AG  parallel  to  OA'  and  SGH  perpendicular  to  OA".  Since 
triangles  OFC  and  AGS  are  equal,  AG  =  c,  GS  =  d.  Hence  the  coor- 
dinates of  the  point  aS  are  OH  ^  a  -\-  c  and  HS  =  h  -{-  d.  The  sum  of 
the  vectors  from  0  to  the  points  (a,  h)  and  {c,  d)  is  the  vector  from  0  to  the 
point  (a  -\-  c,  b  -\-  d),  whose  coordinates  are  the  sums  of  the  corresponding 
coordinates  of  the  two  points. 

Subtraction  of  vectors  is  defined  as  the  operation  inverse  to  addition  of 
vectors.  If  OA  and  OS  are  given  vectors,  the  vcK'tor  OC  for  which  OA 
-\-  OC  =  OS  is  denoted  l)y  OS  -  OA,  and  is  determined  by  the  side  OC 
of  the  parallelogram  with  the  diagonal  OS  and  side  OA. 

18 


§2] 


COMPLEX   NUMBERS 


19 


2.1  Multiplication  of  Vectors.  Let  A  be  a  point  \r,6l  with  the  polar 
codrdinatcs  r,  6.  Then  r  is  the  positive  number  giving  the  length  of  the 
line  OA,  while  6  is  the  measure  of  the  angle  XOA  when  measured  counter- 
clockwise from  OX,  as  in  Trigonometry.  Let  C  be  the  point  \r',  d'l  with 
the  polar  coordinates  r',  6'. 

The  product  OA  •  OC  of  the  vectors  from  0  to  A  =  \r,  6}  and  to  C  = 
Ir',  e'l  is  defined  to  be  the  vector  from  0  to  P  =  \rr',  d  -{■  d'L 


Fig.  11 


To  construct  this  product  geometrically,  let  U  be  the  point  on  the 
X-axis  one  unit  to  the  right  of  0.  Let  the  triangle  OCP  be  constructed 
similar  to  triangle  OUA,  such  that  corresponding  sides  are  OC  and  OU, 
CP  and  UA,  OP  and  OA,  and  such  that  the  vertices  0,  C,  P  are  in  the 
same  order  (clockwise  or  counter-clockwise)  as  the  corresponding  vertices 
0,  U,  A.  Then  OP  :  r'  =  r  :  1,  so  that  the  length  of  OP  is  rr'.  The 
angle  XOP,  measured  counter-clockwise  from  OX,  equals  d  +  6',  and  may 
exceed  four  right  angles.  Hence  the  product  of  the  vectors  OA  and  OC 
is  the  vector  OP. 

If  OC  =  OU,  then  OP  =  OA,  and  Ot/  •  OA  =  OA.  Hence  vector  OU 
plays  the  role  of  unity  in  the  multiplication  of  vectors. 

Division  of  vectors  is  defined  as  the  operation  inverse  to  multiplication 
of  vectors.  If  OA  and  OP  are  given  vectors,  the  vector  OC  for  which 
OA-OC  =  OP  is  denoted  by  OP/OA.  If  A  =  \r,d\  smdP  =  In,  d^]  then 
C  =  \ri/r,  di—  61 .  Division  except  by  zero  is  therefore  always  possible 
and  unique. 

EXERCISES  t 

Lt  Vector  addition  is  associative:    {OA  +  OC)  +  OL  =  OA  +  (OC  +  OL). 
2.t   Vector  multiplication  is  associative:    {OA  •  OC)  •  OL  =  OA  >  {OC  •  OL). 
3.t   Draw  the  figure  corresponding  to  Fig.  12,  when  OA  is  in  the  third  quadrant 
and  OC  in  the  first  quadrant. 


20  THEORY  OF  EQUATIONS  [Ch.  II 

S.f  Symbol  for  Vectors  from  0.  We  consider  only  vectors  starting 
from  the  fixed  point  0.  Such  a  vector  OA  is  uniquely  determined  by  its 
terminal  point  A  =  (a,  h)  and  hence  by  the  Cartesian  coordinates  a,  b  of 
the  point  A  referred  to  fixed  rectangular  axes  OX  and  OY.  We  may 
therefore  denote  the  vector  OA  by  the  symbol  [a,  b].     Then 

(1)  [a,  b]  =  [c,  d]  if  and  only  if  a  =  c,  b  =  d. 

By  the  definition  of  addition  and  subtraction  of  vectors  (§  1), 

(2)  [a,  b]  +  [c,  d]  =  [a  +  c,  6  +  d], 

(3)  [a,  b]  -  [c,  d]  =  [a-c,b-  d]. 

As  our  definition  of  the  product  of  two  vectors  was  made  in  terms  of 
polar  coordinates,  we  must  now  express  the  product  in  terms  of  Cartesian 
coordinates.     By  Fig.  11,  we  have 

a  —  r  cos  6,       b  —  r  sin  6. 

Similarly,  if  the  point  (c,  d)  has  the  polar  coordinates  r',  Q', 

c  =  r'  cos  6' ,     d  =  r'  sin  d'. 
Hence  the  definition  (§2)  of  the  product  of  two  vectors  gives 
[a,  b]  [c,  d\  =  [rr'  cos  (6  +  d'),  rr'  sin  {B  +  6')], 

the  final  numbers  being  the  Cartesian  coordinates  of  the  point  with  the 
polar  coordinates  rr'  and  ^  +  0'.     But 

rr'  cos  {d  +  e')  =  rr'  (cos  6  cos  d'  -  sin  d  sin  d')  ^  ac  -  bd, 
rr'  sin  {6  +  6')  =  rr'  (sin  6  cos  9'  +  cos  d  sin  d')  =  be  +  ad. 

Hence,  finally, 

(4)  [a,  b]  [c,  d]  =  [ac  -  bd,  ad  +  be]. 

Given  a,  b,  e,  f,  we  can  find  solutions  c,  d  of  the  equations 
ac  —  bd  =  e,     ad  -\-  be  ^  f, 

provided  a-  -\- b"^  9^  0,  viz.,  a  and  b  are  not  both  zero.     Then 

[a,b][c,d]  =  [e,f] 

determines  [c,  d],  its  expression  being 

,,.  k/]  ^  Vae  +  bf        af  -  bcl 

^^^  [a,b]      La'  +  ^"       a^  +  fe^J 

Hence  division,  except  by  the  zero  vector  [0,  0],  is  always  possible  and 

unique. 


§4]  COMPLEX  NUMBERS  21 

4.t  Introduction  of  Complex  Numbers.  Giving  up  the  concrete  in- 
terpretation in  §  3  of  the  symbol  [x,  y]  as  the  vector  from  the  origin  to 
the  point  (x,  y),  we  shall  now  think  abstractly  of  a  system  of  elements 
[x,  y]  each  determined  by  two  real  numbers  x,  y,  and  such  that  the  sys- 
tem contains  an  element  corresponding  to  any  pair  of  real  numbers. 
While  the  present  abstract  discussion  is  logically  independent  of  the 
earlier  exposition  of  vectors,  yet  we  shall  be  guided  in  our  present  choice 
of  definitions  of  addition,  multiplication,  etc.,  of  our  abstract  symbols 
[x,  y]  by  the  desire  that  the  vector  system  shall  furnish  us  a  concrete 
representation  of  the  present  abstract  system.  Accordingly,  we  define 
equality,  addition,  subtraction,  multiplication  and  division  of  two  ab- 
stract elements  [x,  y]  by  formulas  (l)-(5).     In  particular,  we  have 

[a,  0]  ±  [c,  0]  =  [a  ±  c,  0], 

[a,  0]  [c,  0]  =  [ac,  0],     -^  =  [^-0]' 

provided  a  5^  0  in  the  last  relation.  Hence  the  elements  [x,  0]  combine 
under  our  addition,  multiplication,  etc.,  exactly  as  the  real  numbers  x 
combine  under  ordinary  addition,  multiplication,  etc.  We  shall  there- 
fore introduce  no  contradiction  if  we  now  impose  upon  our  abstract 
system  of  elements  [x,  y],  subject  to  relations  (l)-(5),  the  further  condi- 
tion that  the  element  [x,  0]  shall  be  the  real  number  x.     Then,  by  (4), 

[0,1]  [0,1]  =  [-1,0]=  -1. 

We  write  i  for  [0,  1].     Hence  i^  =  —1.     Then 

[x,  y]  =  [x,  0]  +  [0,  ij]=x-{-  [y,  0]  [0,  l]=x  +  yi. 

The  resulting  symbol  x  +  yi  is  called  a  complex  number.  For  y  =  0,  it 
reduces  to  the  real  number  x.  For  y  ^  0,  it  is  also  called  an  imaginary 
number.  The  latter  is  not  to  be  thought  of  as  unreal  in  the  sense  that 
its  use  is  illogical.  On  the  contrary,  x  +  yi  is  a  convenient  analytic  rep- 
resentation of  the  vector  from  the  origin  to  the  point  (x,  y),  and  the  sum, 
product,  etc.,  defined  above,  of  two  such  complex  numbers  then  repre- 
sent those  simple  combinations  of  the  two  corresponding  vectors  (§§  1,  2) 
which  are  constantly  used  in  the  applications  of  vectors  in  mechanics  and 
physics.  Since  these  vectors  from  0  are  uniquely  determined  by  their  termi- 
nal points,  we  obtain  a  representation  (§8)  of  complex  numbers  by  points 


22  THEORY  OF  EQUATIONS  [Ch.  ll 

ill  a  piano,  a  representation  of  great  importance  in  mathematics  and  its 
applications. 

If  in  (l)-(5),  we  replace  the  symbol  [a,  h]  by  a  +  bi,  etc.,  we  obtain  the 
formulas  given  in  §  5. 

5.  Formal  Algebraic  Definition  of  Complex  Numbers.  The  equa- 
tion x^  —  —  4:  has  no  real  root,  but  is  said  to  have  the  two  imaginary  roots 
V  —  4  and  —  V  — 4.  We  shall  denote  these  roots  by  2  i  and  —2  i,  agree- 
ing that  I  is  a  definite  number  for  which  i~  =—1.  Similarl}',  we  shall 
write  Vs  i  in  preference  to  V  — 3.  If  p  is  positive,  Vp  is  used  to  denote 
the  positive  square  root  of  p. 

If  a  and  b  are  any  two  real  numbers,  a  +  bi  is  called  a  complex  number 
and  a  —  bi  its  conjugate.  Two  complex  numbers  a  -\-  bi  and  c  +  di  are 
called  equal  if  and  only  if  a  =  c,  6  =  d.  Thus  a  +  &i  =  0  if  and  only  if 
a  =  6  =  0. 

Addition  of  complex  numbers  is  defined  hj 

(a  +  bi)  +  (c  +  di)  =  (a  +  c)  +(6  +  d)i. 

The  inverse  operation,  called  subtraction,  consists  in  finding  a  complex 
number  z  such  that  (c  +  di)  -\-  z  =  a  -\-  bi.     In  notation  and  value,  z  is 

(a  +  bi)  -  (c  +  di)  =  (a  -  c)  +  (6  -  d)i. 
Multiplication  is  defined  by 

(a  +  bi){c  +  di)  =  {ac  -  bd)  +  (ad  +  bc)i, 

and  hence  is  performed  as  in  formal  algebra  with  a  subsequent  reduction 
by  use  of  i^  =  —  1.  If  we  replace  6  by  —6  and  d  by  —d,  the  right  member 
is  replaced  by  its  conjugate.  Hence  the  product  of  the  conjugates  of  two 
complex  members  equals  the  conjugate  of  their  product. 

Division  is  defined  as  the  operation  inverse  to  multiplication,  and  con- 
sists in  finding  a  complex  number  q  such  that  (a  +  bi)q  =  e  -\- fi.  Mul- 
tiplying each  member  by  a  —  bi,  we  find  that  q  is,  in  notation  and  value, 

e  +fi  ^  (e  +/0(a  -  bi)  ^  ae  +  bf      af  -  be  . 
a  +  bi  a2  +  62  a'-^  +  fe'       a'  +  &'^" 

Since  a-  +  6-  =  0  implies  a  =  b  =  0  when  a  and  b  are  real,  division  except 
by  zero  is  possible  and  uni(iue. 


§  6,  71  COMPLEX  NUMBERS  23 

6.  The  Cube  Roots  of  Unity.     The  roots  of  x^  =  1  are  unity  and  the 
numbers  for  which 

^^^a:2  +  a;  +  l  =0,     {x-\-^y=-l     x  +  h  =  ±^VZi. 
X  —  1 

Hence  the  three  cube  roots  of  unity  are  1  and 

EXERCISES 

1.  Verify  that  co'  =  u-,  wcc'  =  1,  w-  +  co  +  1  =  0,  w^  =  1. 

2.  The  sum  and  product  of  two  conjugate  complex  numbers  are  real. 

3.  Express  as  complex  numbers 

3  +  5i        a  +  bi        3  +  V^ 
2-3i'       a-bi'       2  +  V^' 

4.  If  X,  y,  z  are  anj^  complex  numbers, 

x-\ry  =  y  +  X,    {x  +  2/)  +  2  =  .T  +  (y  +  2), 
xy  =  yx,     {xy)z  =  x{yz),     x{y  -{- z)  =  xy  +  xz. 

What  is  the  name  of  the  property  indicated  by  each  equation? 

5.  If  the  product  of  two  complex  numbers  is  zero,  one  of  them  is  zero. 
G-t   Deduce  the  laws  in  §  5  from  those  in  §  4. 

7.  Square  Roots  of  a  +  hi  found  Algebraically.     Given  the  real  num- 
bers a  and  h,  b  9^  0,  Ave  seek  real  numbers  x  and  y  such  that 

a  -\-bi  =  {x  -\-  yi)"  =  a;-  —  7/-  +  2  xyi. 
Thus 

x^  —  y-  ^  a,     2xy  =  h, 

(x2  +  ,/)2  =  (:,2  _  yoy  +  4  ^2y2  =  «2  +  52. 

Since  x  and  y  are  to  be  real  and  hence  x~  +  y'  positive, 

x~  +  ?/2  =  Va^  +  b-, 

the  positive  square  root  being  the  one  taken.     Combining  this  equation 
with  x^  —  y^  =  a,  we  get 

,      Va"  +  62  _^  a  ,      Va^  -\- ¥  -  a 

a;2    =    .,  y2    = 


24  THEORY  OF  EQUATIONS  [Ch.  ll 

Since  these  expressions  are  positive,  real  values  of  x  and  y  may  be  found. 
The  two  pairs  x,  y  for  which  2  xy  =  b  give  the  desired  two  complex  num- 
bers X  +  yi. 

It  is  not  possible  to  find  the  cube  roots  of  a  general  complex  number  by 
a  similar  algebraic  process  (Ch.  Ill,  §  6). 

EXERCISES 

Express  as  complex  numbers  the  square  roots  of 

1.    -7  +  24i.  2.    -11  +  60/. 3.   5-12i. 

4.   4cd+(2c2- 2(^2)1.  5.   c2  -  ^2  _  2  V- c^d^. 

8.  Geometrical  Representation  of  Complex  Numbers.  Using  rec- 
tangular axes  of  coordinates,  we  represent*  a  +  bi  by  the  point  A  =  (a,  6). 
The  positive  number  r  =  Va-  +  b^  giving  the  length  of  OA  is  called  the 
modulus  (or  absolute  value)  of  a  -{-  bi  (Fig.  11).  The  angle  6  =  XOA, 
measured  counter-clockwise  from  OA",  is  called  the  amplitude  (or  argument) 
of  a  +  bi.     Thus 

(6)  a-\-bi  =  r(cos  6  +  i  sin  9). 

The  second  member  is  called  the  trigonometric  form  of  a+  bi. 

If  c  +  di  is  represented  by  the  point  C,  then  the  sum  of  a  +  bi  and 
c -{- di  is  the  complex  number  represented  by  the  point  *S  (Fig.  10) 
determined  by  the  parallelogram  OASC.  Since  OS  =  OA  -\-  AS,  the 
modulus  of  the  sum  of  two  complex  numbers  is  equal  to  or  less  than  the  sum 
of  their  moduli. 

For  example,  the  cube  roots  of  unity  are  1  and 


I 


=  cos  120°  +  i  sin  120°, 


0^=  -i-iV3i 


=  cos  240°  +  i  sin  240°, 


Fig.  13 


and  are  respresented  by  the  points  marked  1,  w,  w-  in  Fig.  13.     They  form 
*  It  wiU  be  obvious  to  the  reader  who  has  not  omitted  §§  1-4  that  the  present  rep- 
resentation is  essentially  equivalent  to  the  representation  oi  a  -\-  bi  by  the  vector  from 
O  to  the  point  (o,  6). 


§  9. 10]  COMPLEX  NUMBERS  25 

the  vertices  of  an  equilateral  triangle  inscribed  in  a  circle  of  unit  radius 
and  center  at  the  origin  0. 

9.  The  product  of  the  complex  number  (6)  by  r'(cos  a  -{-  i  sin  a)  is 

rr'  [cos  (0  +  a)  +  i  sin  {6  +  «)  ], 
since 

(7)  (cos  6  -\-  i  sin  9) {cos  a  -{-  i  sin  a)  =  cos  (6  -{-  a)  -^  i  sin  {d  -{-  a). 
The  latter  follows  from 

cos  9  cos  a  —  sin  9  sin  a  =  cos  {9  -\-  a), 
cos  0  sin  a  +  sin  9  cos  a  =  sin  {9  -^  a). 

Hence  the  modulus  of  the  product  of  two  complex  numbers  equals  the  product 
of  their  moduli,  and  the  amplitude  of  the  product  equals  the  sum  of  their 
amplitudes. 

The  product  may  be  found  geometrically  as  in  Fig.  12. 

For  the  special  case  a  =  9,  (7)  becomes 

(cos  9  -\-  i  sin  6)-  =  cos  2  0  +  t  sin  2  9. 

This  is  the  case  n  =  2  of  formula  (8).  In  particular,  we  see  why  the 
amplitude  of  w^  is  240°  when  that  of  co  is  120°  (end  of  §  8). 

10.  De  Moivre's  Theorem.     //  n  is  any  positive  integer, 

(8)  (cos  9  -\-i  sm.9y  =  cos  n9  +  i  sin  n9. 

This  relation  is  an  identity  if  n  =  1  and  was  seen  to  hold  if  n  =  2. 
To  proceed  by  mathematical  induction,  let  it  be  true  if  n  =  m.  Using 
(7)  for  a  =  m9,  we  then  have 

(cos  0  +  1  sin  0)'"+^  =  (cos  9  -\- i  sin  6)  (cos  6  +  i  sin  9Y 

=  (cos  0  +  i  sin  9)  {cos  m9  +  i  sin  m9)  =  cos  (m  +  1)0  +  i  sin  (m  -{-1)9. 

Hence  (8)  is  true  also  if  /i  =  m  +  1.     The  induction  is  thus  complete. 

Since  cos  9  -{-  i  sin  e  represents  the  vector  from  the  origin  0  to  the  point  ll,  ol, 
given  in  polar  coordinates,  its  nth  power  represents  (§2)  the  vector  from  0  to  the 
point  1 1,  ndl  and  hence  is  cos  nd  +  i  sin  nd. 


26  THEORY  OF  EQUATIONS  [Ch.  ll 

11.  Cube  Roots.  To  find  the  cube  roots  of  a  complex  number,  we 
first  express  it  in  the  trigonometric  form  (6).     For  example, 

4  a/2  +  4  V2  i  =  8  (cos  45°  +  i  sin  45°). 
If  it  has  a  cube  root  of  the  form  (6),  then,  by  (8), 

r^  (cos Se  +  isinSd)  =8  (cos 45°  +  i sin 45°). 

Their  moduli  r^  and  8  must  be  equal,  so  that  the  positive  real  number  r 
equals  2.  Since  3  6  and  45°  have  equal  cosines  and  equal  sines,  they  differ 
by  an  integral  multiple  of  360°.     Thus 

6  =  lo°  +  k'  120°  (k  an  integer). 

Since  in  (6)  we  may  replace  dhy  6  -^  360°  without  changing  a  +  hi,  we  ob- 
tain just  three  distinct  cube  roots  (given  hy  k  =  0,  1,  2): 

2  (cos  15°  +  i  sin  15°),  2  (cos  135°  +  i  sin  135°),  2  (cos  255°  +  i  sin  255°). 

EXERCISES 

1.  Verify  that  the  last  two  numbers  equal  the  products  of  the  first  number  by 
w  and  w*,  given  at  the  end  of  §  8. 

2.  Find  the  tliree  cube  roots  of  —27;  those  of  —  i. 

3.  Find  the  three  cube  roots  of  —  §  +  ^  Vs  i. 

12.  nth  Roots.  Let  p  be  a  positive  real  numljcr.  As  illustrated  in 
§  11,  it  is  evident  that  the  nth  roots  of  p  (cos  A  +  i  sin  A)  are  the  prod- 
ucts of  the  7ith  roots  of  cos  A  -{-  i  sin  A  by  the  positive  real  nth  root  of 
p.     Let  an  nth  root  of  cos  A  -}-  i  sin  A  be  of  the  form  (6).     Then,  by  (8), 

r"(cos  nd  +  i  sin  7id)  =  cos  A  -\-  isiriA. 

Thusr"  =  1,  r  =  1,  and  nO  =  A  +  A-  •  360°,  where  k  is  an  integer.  Thus 
n  distinct  nth  roots  of  cos  A  -\-  i  sin  A  are  given  by 

._.  A-\-k-  360°  ,    ■  ■    A+k'  360°      ,,       ^  ,  ., 

(9)  cos h^sm ■      (A:  =  0,  1,  .  .  .  ,  w— 1), 

n  n 

whereas  k  =  n  gives  the  same  root  as  ^  =  0,  and  k  =  7i  +  1  the  same 
root  ask  =  1,  etc.  Hence  any  number  9^  0  has  exactly  n  distinct  wth  com- 
plex roots. 

EXERCISES 

1.  Find  the  five  fifth  roots  of  -  L 

2.  Find  the  nine  ninth  roots  of  L     Which  are  roots  of  .r'  =  1? 

3.  Simplify  the  trigonometric  forms  of  the  four  fourth  roots  of  unity.  Check 
the  result  by  factoring  x''  —  1. 


131 


COMPLEX  NUMBERS 


27 


13.   Roots  of  Unity. 

(10) 


By  (9)  the  n  distinct  nth  roots  of  unity  are 

(/c  =  0,  1,  .  .  .  ,  n-  1), 


2kT  ,    .  .    2  kit 

cos h  I  sin 

n  n 


where  now  the  angles  are  measured  in  radians  (an  angle  of  180  degrees 
equals  tt  radians,  where  tt  =  3.1416,  approximately).  For  k  =  0,  (10) 
reduces  to  1,  which  is  an  evident  nth  root  of  unity.     For  /b  =  1,  (10)  is 


(11) 


27r    ,     .    .     27r 

COS \-  I  sni  — . 

71  n 


By  DeMoivre's  Theorem  (§  10),  the  general  number  (10)  equals  the  A;th 
power  of  r.     Hence  the  n  distinct  nth  roots  of  unity  are 

(12)  r,  r2,  r*,  .  .  .  ,  r"-i,  r"  =  1. 

I  The  n  complex  numbers  (10),  and  therefore  the  numbers  (12),  are  rep- 
resented geometrically  by  the  vertices  of  a  regular  polygon  of  n  sides 
inscribed  in  the  circle  of  radius  unity  and  center  at  the  origin  with  one 
vertex  on  the  a:-axis  (Fig.  14). 


Fig.  14 


Fig.  15 


For  n  =  3,  the  numbers  (12)  are  co,  or,  1,  shown  in  Fig.  13. 

For  n  =  4,  we  have  r  =  cos  7r/2  +  i  sin  t/2  =  i.  The  fourth  roots  of 
unity  (12)  are  i,  i-  =  —  1,  i'  =  —i,i^  =  1.  These  are  represented  by  the 
vertices  of  a  square  inscribed  in  a  circle  of  radius  unity  (Fig.  15). 

EXERCISES 

1.  For  71  =  6,  r  =  —  w^.  The  sixth  roots  of  unity  are  therefore  the  three  cube 
roots  of  unity  and  their  negatives.     Check  by  factoring  x^  —  1. 

2.  From  the  point  representing  a  +  hi  how  do  you  obtain  that  representing 
—  (a  +  hi)l  Hence  derive  from  Fig.  13  and  Ex.  1  the  points  representing  the  six 
sixth  roots  of  unity. 

3.  Which  powers  of  a  ninth  root  (11)  of  unity  are  cube  roots  of  unity? 


28  THEORY  OF  EQUATIONS  ICh.  II 

14.  Primitive  nth  Roots  of  Unity.  An  nth  root  of  unity  is  called 
primitive  if  no  power  of  it,  with  a  positive  integral  exponent  less  than  n, 
equals  unity.  Since  only  the  last  one  of  the  numbers  (12)  equals  unity, 
the  number  r,  given  by  (11),  is  a  primitive  nth  root  of  unity. 

For  n  =  4,  both  i  and  —i  are  primitive  fourth  roots  of  unity,  while 
1  and  —1  are  not.  Just  as  i^  =  —1  and  i*  =  +1  are  not  primitive  fourth 
roots  of  unity,  so  r*  is  not  a  primitive  wth  root  of  miity  if  k  and  n  have  a 
common  divisor  d  {d  >  1).     Indeed, 

n  k 

(r^)d  =  (r")T=  1, 

whereas  n/d  is  a  positive  integer  less  than  n.  But  if  k  and  n  are  relatively 
prime,  i.e.,  have  no  common  divisor  exceeding  unity,  r^'  is  a  i3rimitive  nth 
root  of  unity.  To  prove  this,  we  must  show  that  (r*)'  ?^  1  if  Hs  a  posi- 
tive integer  less  than  n.     Now,  by  De  Moivre's  Theorem, 

2klT    ,     .    .     2klTr 

n'  =  cos- H  I  sin 

71  n 

If  this  were  unity,  2  klir/n  would  be  a  multiple  of  2  r,  and  hence  kl  a 
multiple  of  n.  Since  k  is  relatively  prime  to  n,  the  second  factor  I  would 
be  a  multiple  of  7i,  w^hereas  0  <  I  <  n.  Hence  the  primitive  nth  roots  of 
unity  are  those  of  the  numbers  (12)  whose  exponents  are  relatively  prime  to  w. 

EXERCISES 

1.  The  primitive  cube  roots  of  unity  are  w  and  ur. 

2.  For  r  given  by  (11),  the  primitive  ?/th  roots  of  unity  are  (i)  for  n  =  G,  r,  r^; 
(ii)  for  71  =  12,  r,  r^,  r^,  r". 

3.  For  n  a  prime,  any  7ith  root  of  unity,  other  than  1 ,  is  primitive. 

4.  If  r  is  a  primitive  15th  root  of  unity,  r'',  r^,  r',  ?•'-  are  the  primitive  oth  roots 
of  unity,  and  r^,  r'"'  are  the  primitive  cul)e  roots  of  unity.  Show  tliat  their  8  prod- 
ucts by  pairs  give  all  of  the  primitive  15th  roots  of  unit}'. 

5.  If  n  is  the  product  of  two  primes  p  and  q,  there  are  exactly  (p  —  l){q  —  1) 
primitive  nth  roots  of  unity. 

6.  If  p  is  any  primitive  nth.  root  of  unity,  p,  p-,  p^,  .  .  .  ,  p"  are  distinct  and  give 
all  of  the  ?ith  roots  of  unity.  Of  these,  p'^  is  a  primitive  ?ith  root  of  unity  if  and 
only  if  k  is  relatively  prime  to  7i. 


15.   Imaginary  Roots  Occur  in  Pairs.     The  roots  of  x^  +  2  ex  +  d  =  0 


are 


(13)  -c+Vc'-d,      -c-Vc"-  d. 


§  16]  COMPLEX  NUMBERS  29 

If  c  and  d  are  real,  these  roots  are  both  real  or  are  conjugate  imaginaries. 
The  latter  case  illustrates  the  following 

Theorem.     If  a  and  b  are  real  numbers,  6  5^  0,  and  if  a  -\-  bi  is  a  root 
of  an  equation  with  real  coefficients,  then  a  —  hi  is  a  root. 
Let  the  equation  be  f(x)  =  0.     Divide  /(.r)  by 

(14)  (x  -  a)2  +  62  =  (a:  -  a  -  bi)  (.c  -  a  +  bi) 

until  we  reach  a  remainder  rx  -\-  s  oi  degree  less  than  the  degree  of  the 
divisor  in  x.  Evidently  r  and  s  are  real.  If  the  quotient  is  Q{x),  we 
have 

f(x)  =  Q{x)  1  (x  -  a)2  +  6^1  -\-rx-\-s, 

identically  in  x  (Ex.  6,  p.  15).  Let  x  =  a  -}-  bi.  Since  this  is  a  root  of 
f{x)  =  0,  we  see  that 

0  =  r(a  +  bi)  +  s,     0  =  ra  +  s,     0  =  rb. 

Since  b  ^  0,  we  have  r  =  0  and  then  s  =  0.  Thus/(.r)  has  the  factor  (14), 
so  that/(.r)  =  0  has  the  root  a  —  bi. 

16.  t   Generalization  of  the  theorem  in  §15.     The  sum  of  the  roots 

(13)  oi  x~  -\-  2  ex  -\-  d  =  0  equals  the  negative  of  the  coefficient  2  c  of  a;, 

and  their  product  equals  the  constant  term  d.     It  follows  that  2  +  i  and 

—  2  are  the  roots  of 

z^'  -iz-  4-2  i  =  0, 

and  that  2  —  i  and  —  2  are  the  roots  of 

z^-\-iz-4:-^2i  =  0. 

We  have  here  an  illustration  of  the  following 

Theorem.  If  a  and  b  are  real  nu7nbers  and  if  a  -\-  bi  is  a  root  off(z)  =  0, 
then  a  —  bi  is  a  root  of  g(z)  =  0,  where  g{z)  is  obtained  from  the  -polynomial 
f{z)  by  replacing  each  coefficient  c  +  di  by  its  conjugate  c  —  di. 

Consider  any  term  (c  +  di)z''  of  f(z).  Replace  z  hy  x  -\-  yi,  where  x 
and  y  are  real.     The  term 

{c  +  di)(x  +  yi)^ 

of  f(x  +  yi)  has  as  its  conjugate  imaginary  the  product 

(c  —  di)(x  —  yi)'' 


30  THEORY  OF  EQUATIONti  [Ch.  ii 

of  the  conjugates  of  the  factors  of  that  term  (§  5).     But  the  new  product 
is  a  term  of  g{x  —  yi).     Hence  the  latter  is   the  conjugate  A  —  Bi  of 
j{x  -\-  iji)  ^  A  -\-  Bi,  where  A  and  B  are  polynomials  in  x  and  y  \\ith 
real  coefficients. 
Take  x  =  a,  y  =  h.     Then  A=B  =  0  by  hypothesis.     Hence  g{a—hi)  =  0. 

EXERCISES 

l.f   The  theorem  in  §  15  is  a  corollary  to  that  in  §  16. 

2.  Solve  x^  -  3  .c2  -  6  .c  -  20  =  0,  with  the  root  - 1  +  V^ 

3.  Solve  .r*  -^x^-\-bx'-2x-2  =  0,  with  the  root  1  -  i. 

4.  Find  the  cubic  e(iuation  with  real  coefficients  two  of  whose  roots  are  1  and 
3  +  2i. 

5.t  Given  that  x^  +  (1  —  O-i-'  +  1=0  has  the  root  i,  find  a  cubic  equation 
with  the  root  —i.  Form  an  equation  with  real  coefficients  whose  roots  include 
the  roots  of  these  two  cubic  equations. 

6.  If  an  equation  with  rational  coefficients  has  a  root  a  +  v  6,  where  a  and  h  are 
rational,  but  V6  is  irrational,  it  has  the  root  a  —  \/h.     [Use  the  method  of  §  15.] 

7.  Solve  x^  -  4  x^*  +  4  .r  -  1  =  0,  with  the  root  2  +  Vs. 

8.  Solve  x3  -  (4  +  V3).c2  +  (o  +  4  V3)x  -  5  Vs  =  0,  with  the  root  Vs. 

9.  Solve  the  equation  in  Ex.  8,  given  that  it  has  the  root  2  +  i. 

10.  What  cubic  equation  with  rational  coefficients  has  the  roots  |,  ^  +  V2  ? 


CHAPTER  III 
Algebraic  and  Trigonometric  Solution  of  Cubic  Equations 

1.  Reduced  Cubic  Equation.     If  in  the  general  cubic  equation 

(1)  x^ -\- hx^ -\- ex -\- d  =  0, 

we  set  X  =  y  —  6/3,  we  obtain  a  reduced  cubic  equation 

(2)  y'  +  py  +  q  =  0, 
where 

(3)  P  =  ^-3'       ^  =  ^~3+'27'* 

A  geometrical  interpretation  of  this  process  was  given  in  Ex.  5,  p.  10. 
We  shall  find  the  roots  yi,  tjo,  y^  of  (2).     Then  the  roots  of  (1)  are 

(A\  h  h  h 

(4)  Xi  =  yi-  ^,      X2  =  y2-  ^y      X3  =  yz  -  ^• 

2.  Algebraic  Solution  of  Cubic  Equation  (2).  We  shall  employ  a 
method  essentially  that  given  by  Vieta  *  in  1591.  We  make  the  substi- 
tution 

(5)  y  =  .-l 

in  (2)  and  obtain 

3 


Multiplying  each  member  by  z^,  we  get 

(6)  z'  +  qz'-^  =  0. 
Solving  this  as  a  quadratic  equation  for  2',  we  obtain 

(7)  ^=-|±Vg,     «  =  (fj+(f 

*  Opera  Math.,  IV,  published  by  A.  Anderson,  Paris,  1615. 
31 


32  THEORY  OF  EQUATIONS  [Ch.  ill 

By  Ch.  II,  §  11,  any  number  has  three  cube  roots,  two  of  which  are  the 
products  of  the  remaining  one  by 

(8)  co=-i+iV3z,     a;2=-l-iV3  2. 
Since 

(_|  +  VS)(-|-VS)=(-|J. 

we  can  choose  particular  cube  roots 

(9)  A  =  \/-^+VR,      b  =  \/-'^-VR, 

such  that  AB  =  —  p/3.     Then  the  six  values  of  z  are 

A,  01  A,  0)^ A,  B,  a:B,  co25. 

These  can  be  paired  so  that  the  product  of  the  two  in  each  pair  is  —  p/3: 

AB  =  -p/3,      uA  '  orB  =  -p/S,    oi-A  '  uB  =  -p/3. 

Hence  with  any  root  z  is  paired  a  root  equal  to  —  p/{S  z).     By  (5),  the 
sum  of  the  two  is  a  value  of  y.     Thus  the  three  roots  of  (2)  are 

(10)  yi  =  A+B,     y.  =  co.4  +  oi'-B,    y^  =  o>-A  +  co5. 

These  are  known  as  Cardan's  formulce  for  the  roots  of  a  reduced  cubic 
equation  (2).  The  expression  ^4.  +  5  for  a  root  was  first  published  by 
Cardan  in  his  Ars  Magna,  1545,  although  he  had  obtained  it  from  Tartaglia 
under  promise  of  secrecy. 

EXERCISES 

1.  For  ?/  -loy-  126  =  0,    y  =  z  +  o/z    and 

28  -  1262^  +  125  =  0,        2^  =  1  or  125,        2  =  1,  co,  0,2,  5,  5  «,  5  «'. 

The  first  three  2's  give  the  distinct  7/'s :  G,  to  +  5  co-,  oj-  +  5  w. 

2.  Solve  1/  -  18  2/  +  35  =  0.  3.   Solve  x^  +  6  x^  +  3  x  +  IS  =  0. 
4.  Solve  y^-2y  +  4:  =  0.  5.   Solve  28  x'  +  9  x^  -  1  -  0. 

6.  Using  w^  +  o)  +  1  =  0,  show  from  (10)  that 

yi  +  2/2  +  j/3  =  0,        7/i?/2  +  2/12/3  +  2/22/3  =  p,        2/12/22/3  =  —q. 

7.  By  (3),  (4)  and  Ex.  6,  show  that,  for  the  roots  of  (1), 

Xi  +  X2  +  X3=  —b,        X1X2  +  X1X3  +  X2X3  =  c,        X1X2X3  =  — d 


§31  CUBIC  EQUATIONS  33 

3.  Discriminant.     By  (10)  and  w^  =  1, 

yi-y2=  (i-co)(A-co25), 

yi-yz=  -co^  (1  -  a))(A  -  coB), 

?/2  -  ?/3   =  CO  (1   -  C0)(A   -  B). 

To  form  the  product  of  these,  note  that  co^  =  1  and,  by  (8), 

(1  -  co)3  =  3  (co2  -  co)  =   -3  VS  Z. 

Since  the  cube  roots  of  unity  are  1,  w,  co^,  we  have 

x^  —  1  =  (x  —  l)(.r  —  aj)(x  —  co^), 
identically  in  x.     Taking  x  =  A/B,  we  see  that 

(11)  A'-B^=  (A  -  B){A  -  co5)(A  -  co25). 
The  left  member  equals  2  VTi  by  (9).     Hence 

(12)  (2/1  -  y^)  (yi  -  2/3)  (2/2  -ys)=QV3VR  i. 

The  product  of  tlie  squares  of  the  differences  of  the  roots  of  any  equation 
in  which  the  coefficient  of  the  highest  power  of  the  variable  is  unity  shall  be 
called  the  discriminant  of  the  equation.  Thus  the  discriminant  is  zero  if 
and  only  if  two  roots  are  equal,  and  is  positive  if  all  the  roots  are  real. 

In  view  of  (12)  the  discriminant  A  of  the  reduced  cubic  equation  (2) 
has  the  value 

(13)  ^  A  =  -108/^  =  -4793-2752. 

By  (4),  Xi  —  X2  =  yi  —  yo,  etc.  Hence  the  discriminant  of  the  general 
cubic  (1)  equals  the  discriminant  of  the  corresponding  reduced  cubic  (2). 
By  (3)  and  (13), 

(14)  ^  =  18  bed  -4:¥d-{-  b^c^  -  4  c^  -27  d\ 
It  is  sometimes  convenient  to  employ  a  cubic  equation 

(15)  ax^  -}-  bx^  +  C.T  +  d  =  0, 

in  which  the  coefficient  of  x^  has  not  been  made  unity  by  division.  The 
product  P  of  the  squares  of  the  differences  of  its  roots  is  evidently  derived 
from  (14)  by  replacing  6,  c,  d  by  b/a,  c/a,  d/a.     Thus 

(16)  a^P  -  18  abed  -  4:¥d -\-  fcV  -  4  ac^  -  27  aW. 

This  expression  (and  not  P  itself)  is  called  the  discriminant  *  of  (15). 

*  Some  writers  define  —  ^V  «*^  to  be  the  discriminant  of  (15)  and  hence  —2V  ^  as 
tkat  of  (1).    On  this  point  see  Ch.  IV,  §  4. 


34  THEORY  OF  EQUATIONS  (CH.m 

4.  Theorem.  A  cubic  equation  with  real  coefficients  has  three  distinct 
real  roots,  a  single  real  root,  or  at  least  two  equal  real  roots,  according  as  its 
discriminant  is  positive,  negative  or  zero. 

It  suffices  to  prove  the  theorem  for  a  reduced  cubic  equation  (2)  in 
which  p  and  q  are  real.  First,  let  A  =  0.  By  (13),  R  =  0.  Using  (8), 
we  find  that  the  roots  (10)  are 

(17)  A+B,   -HA-{-B)±UA-B)V3i. 

But  .4  and  B,  in  (9),  may  now  be  taken  to  be  real,  since  R  =  0. 

li  R  >  0,  A  9^  B  and  A  +  B  is  the  only  real  root.  U  R  =  0,  then 
A  =  B  and  the  roots  are  real  and  at  least  two  are  equal. 

Next,  let  A  >  0,  so  that  R  <0.  Since  —lq+  ^/R  is  an  imaginary  num- 
ber it  has  (Ch.  II,  §  11)  a  cube  root  of  the  form  A  =  a  -]-  jSi,  where  a  and 
13  are  real  and  /3  ?^  0.  Then  (Ch.  II,  §  16)  5  =  a  —  ^i  is  a  cube  root  of 
—  2?  ~  \^R.  For  these  cube  roots,  the  product  AB  is  real  and  hence 
equals  —  p/3,  as  required  in  §  2.     Hence 

2/1  =  2  a,     y.2=  -a-  ^VS,     2/3  =  -a  +  /3  VS. 
These  real  roots  are  distinct  since  A  f^  0. 


EXERCISES 

Find  by  means  of  A  the  number  of  real  roots  of 

1.  ?/3  -  15 y  +  4  =  0.    2.   7f  -  27 y  +  54  =  0.    3.  x^  +  Ax^  -  llx  +  6  =  0. 

4.  Using  A  =  (.Ti  —  XiY  (xi  —  XsY  {x^  —  XsY,  show  that,  if  .ri  and  .T2  are  con- 
jugate imaginaries  and  hence  X3  real,  A  <  0;  if  the  x's  are  all  real  and  distinct, 
A  >  0.     Deduce  the  theorem  of  §  4. 

5.  Deduce  the  same  theorem  from  Ch.  I,  §  9. 

5.  Irreducible  Case.  When  the  roots  of  a  cubic  equation  are  all  real 
and  cUstinct,  R  is  negative  (§4),  so  that  Cardan's  formulae  present  their 
values  in  a  form  involving  cube  roots  of  imaginaries.  This  is  called  the 
irreducible  case.*  We  shall  derive  modified  formulae  suitable  for  numer- 
ical work.  Since  any  complex  number  can  be  expressed  in  the  trigono- 
metric form,  we  can  find  r  and  d  such  that 

(18)  -^,q-\-VR  =  r  (cos d  +  i sin 6). 

*  This  term  is  not  to  be  confuted  with  "irreducible  equation." 


§61  CUBIC  EQUATIONS  35 

In  fact,  the  conditions  for  this  equahty  are 


Hence 


(19) 


—  1  g  =  r  COS  6,     R  =  —r^  sin^  9. 
J.2  =  J.2  (cos2 d  +  sin2 d)  =12^  -  R  =  ^, 


=  V/^'    -^  =  i^-\/i? 


Since  i2  is  negative,  p  is  negative  and  r  is  real.     Since  R  <  0,  the  value 

(19)  of  cos  d  is  numerically  less  than  unity.     Hence  d  can  be  found  from 
a  table  of  cosines. 

The  complex  number  conjugate  to  (18)  is 

(20)  - 1  g  -  V^  =  ^(cos  d  -  i  sin  6). 
The  cube  roots  of  (18)  and  (20)  are 

d  +  m-  360'^ _^  .  .    e  +  m-  360°]  ,         ^    .    _, 
5 it  I  sm 5 (m  =  0,  1,2). 


-^[cos 


For  a  fixed  value  of  m  the  product  of  these  two  numbers  is  —  p/3.  Hence 
their  sum  is  a  root  of  our  cubic  equation.  Thus  if  R  is  negative,  the  three 
distinct  real  roots  are  

(21)  2Y-^cos ^ (m  =  0,  1,  2). 

EXERCISES 

1.  Solve  the  cubics  in  Exs.  1,  2,  page  34. 

2.  Solve  ?/  -  27/  -  1  =  0.  3.  Solveif-  7y  +  7  =  0. 
4.   Find  constants  r  and  s  such  that 

y^  +  py  +  q  =  tzt  ^^  (2/  +  '^^^  -  s(y  +  rfl 

identically  in  y.     Hence  solve  the  reduced  cubic  equation. 

6.t  Algebraic  Discussion  of  the  Irreducible  Case.  Avoiding  the  use 
of  trigonometric  functions,  we  shall  attempt  to  find  algebraically  an 
exact  cube  root  x  -\-  yi  oi  a  -{-  hi,  where  a  and  b  are  given  real  numbers, 
b  7^  0.     We  desire  real  numbers  x  and  y  such  that 

(x  +  yiy  =  a  +  bi, 


36  THEORY  OF  EQUATIONS  [Ch.  Ill 

whence  a;^  —  3  xy-  =  a,     3  x^y  —  y^  =  b. 

Thus  2/  5^  0  and  we  may  therefore  set  x  =  sy.     Hence 

{s^-Ss)y'  =  a,     (3s2-l)?/3  =  6 

Eliminating  y^,  we  get 

0  0 

Set  s  =  t  -\-  a/b.     We  obtain  the  reduced  cubic  equation 

The  R  of  (7)  is  here  —  k-.     Thus  Cardan's  formute  for  the  roots  i  involve 


A  =  \/~  k  +  ki  =  \/^ .  </a  +  6i. 

While  the  first  factor  is  the  cube  root  of  a  real  number,  the  second  is 
exactly  the  cube  root  which  we  started  out  to  find. 

Hence  this  algebraic  process  in  conjunction  with  that  in  §  2  fails  to  give 
us  the  real  roots  of  our  cubic  equation.  Conceivabl}^  other  algebraic  proc- 
esses would  succeed;  but  it  can  be  proved  *  rigorously  that  a  cubic  equation 
with  rational  coefficients  having  no  rational  root,  but  having  three  real 
roots,  cannot  be  solved  in  terms  of  real  radicals  only.  Hence  there  does 
not  exist  an  algebraic  process  for  finding  the  real  values  of  the  roots  in 
the  irreducible  case. 

A  cube  root  of  a  general  complex  number  cannot  be  ex-pressed  in  the 
form  X  +  7ji,  where  x  and  y  involve  only  real  radicals.  For,  if  so,  Cardan's 
formula)  could  be  simplified  so  as  to  express  the  roots  of  any  cubic  equa- 
tion in  terms  of  real  radicals  only. 

7.  t  Trigonometric  Solution  of  a  Cubic  Equation  with  A  >  0.  In  the 
irreducible  case  we  may  avoid  Cardan's  formula^  and  the  simplifications 
in  §  5.  The  same  final  results  are  now  obtained  by  a  direct  solution  based 
upon  the  well-known  trigonometric  identity 

cos  3  a:  =  4  cos^  x  —  S  cos  x. 

*  H.  Weber  and  J.  Wellstein,  Encyklopddie  der  Elementar-Mathemalik,  I,  cd.  1,  p.  325; 
ed.  2,  p.  373;  ed.  3,  p.  364. 


§71  CUBIC  EQUATIONS  37 

This  may  be  written  in  the  form 

^  —  I  z  —  lcos3x  =  0  {z  =  cos  x). 

To  transform  cubic  (2)  into  this  one,  set  y  =  nz.    Thus 

n^        nr 
The  two  cubic  equations  are  identical  if 

—  n3 


=  \/4^.     eos3x  =  :i«.^ 


2     ■    V    2f 


Since  /2  <  0,  p  <  0  and  the  value  of  cos  3  a:  is  real  and  numerically  <  1. 
Hence  we  can  find  3  x  from  a  table  of  cosines.  The  three  values  of  z  are 
then 

cos  X,     cos  {x  +  120°),     cos  {x  +  240°). 

Multiplying  these  by  n,  we  get  the  three  roots  y. 
Example.     For  if  —  2y  —  1  =  Q,  we  have 

n2  -  8/3,     cos  3  X  -  V27/32,     3  x  =  23°  17'  0", 
cosx  =  0.99084,     cos  (x  +  120°)  =  -0.61237,    cos  {x  +  240°)  =  -0.37847, 
y  =  1.61804,  -  1,  -  0.61804. 

EXERCISESt 

Solve  by  the  last  method 

1.  ?/  -  7  2/  +  7  -  0.  2.   x3  +  3  x2  -  2  X  -  5  =  0. 

3.  x3  4-  .x2  -  2  X  -  1  =  0.  4.   x3  +  4  x2  -  7  =  0. 

5.  The  cubic  for  Mn  §  6  has  three  real  roots;   in  just  three  of  the  nine  sets  of 
solutions  X,  y,  both  are  real. 


CHAPTER  IV 
Algebraic  Solution  of  Quartic  Equations 

1.   Ferrari's  Method.     Writing  the  quartic  equation 

(1)  x"  +  bx^  +  cx^ -\- dx  +  e  =  0 
in  the  equivalent  form 

(x''  +  i  bxy  =  (i  ^2  _  c).^2  -dx-e 

and  adding  {x-  +  |  bx)2j  +  I  ?/^  to  each  member,  we  get 

(2)  (x2  +  Pa;  +  i  yY  ={lb'~-c  +  y)x'  +  {\hy  -  d)x  +  |  y^  _  g. 

We  seek  a  value  yx  of  y  such  that  the  second  member  of  (2)  shall  be  the 
square  of  a  linear  function  of  x.     For  brevity,  write 

(3)  ¥-4.c  +  4:y,  =  t\ 

We  here  assume  that  t  f^  0  (c/.  Exs.  3,  4,  p,  40).     We  therefore  desire 
that 

(4)  i  ex'  +  (1  by,  -d)x^-\  yi^  -  e  =  (|  te  +  iMnl^l 
The  condition  for  this  is  that  the  terms  free  of  x  be  equal: 


6^  —  4  c  +  4  ^1 

Hence  yi  must  be  a  root  of  the  resolvent  cubic  equation 

(6)  ?/  -  ci/2  -\.  (bd-  4.  e)y  -  6^6  +  4  ce  -  d^  =  o. 

After  finding  (Ch.  HI)  a  root  yi  of  this  cubic  equation,  we  can  easily 
get  the  roots  of  the  quartic  equation.  In  view  of  (2)  and  (4),  each  root 
of  the  quartic  equation  satisfies  one  of  the  quadratic  equations 

,.  [x-'  +  lib-  t)x  +  hy,-  {h  by,  -  d)'t  =  0, 

^  ^  U-  +  Hb  +  t)x-{-hyi  +  {h  by,  -  d)/t  =  0. 

38 


§2.31  QUART  I C  EQUATIONS  39 

EXERCISES 

1.  For  x^  +  2  x'  -  12  a;2  -  10  X  +  3  =  0,  show  that  (6)  becomes 

2/^  +  12  2/2  —  32  ?/  —  256  =  0,  with  the  root  iji  =  —4,   and  that  (7)  then  become 

x2  +  4  X  -  1  =  0,     x2  -  2  X  -  3  =  0, 

with  the  roots  —2  ±  V5;    3,  —1. 

2.  Solve  x^  -  2  x3  -  7  x2  +  8  X  +  12  =  0. 

3.  Solve  x^  -  8  x^  +  9  x2  +  8  X  -  10  =  0. 

2.   Relations  between  the  Roots  and  Coefficients.     Let  Xi  and  X2  be 

the  roots  of  the  first  quadratic  equation  (7),  Xs  and  Xi  those  of  the  second. 
The  sum  and  product  of  the  roots  of  x^  -{-  Ix  +  m  =  0  are  —I  and  m 
respectively  (Ch.  II,  §  16,  or  Ch.  VI,  §  1).     Hence 

.-^1  +  0:2=  -hQj  -  t\,     X1X2  =  1 2/1  -  (^  hyi  -  d)/t, 


(8) 

\xz-\-  Xi=  - \  (6  +  0;    ^32:4  =  I  yi  +  {\  hiji  -d)/L 

Using  also  (5),  we  find  at  once  that 

(9)  xi -j-  Xi  +  X3  +  Xi  =  -b,    XxXiXzXi  =  \y^  - (|  y^  -e)  =  e, 

(10)  X1X2  +  a;irr3  +  2:1X4  +  0:2X3  +  X2X4 + X3.T4  =  X1.T2  +  (xi + X2)  (xs + X4)  +  X3X4  =  c, 

(11)  X1X2X3  +  X1X2X4  +  X1X3X4  +  X2X3X4  =  XiX2(x3  +  X4)  +  X3X4(xi  +  X2)  =  —  d. 

It  follows  from  Ex.  3,  p.  40  that  (9)-(ll)  hold  also  when  there  is  no  root 
^1  for  which  t  ^  0. 

For  any  quartic  equation  (1),  the  sum  of  the  roots  is  —b,  the  sum  of  the 
products  of  the  roots  two  at  a  time  is  c,  the  siwi  of  the  products  three  at  a  time 
is  —  d,  the  product  of  all  four  is  e. 

A  proof  based  upon  more  fundamental  principles  is  given  in  Ch.  VI,  §  1. 

3.   Roots  of  the  Resolvent  Cubic  Equation.     These  are 

(12)  yi  =  XiXo  +  X3X4,     2/2  =  X1X3  +  X2X4,     2/3  =  X1X4  +  X2X3. 

The  first  relation  follows  from  (8).  If,  instead  of  yi,  another  root  of  (6) 
be  employed  as  in  §  1,  quadratic  equations  different  from  (7)  are  ob- 
tained, such  however  that  their  four  roots  are  Xi,  X2,  X3,  X4,  paired  in  a  new 
way.  This  leads  us  to  expect  that  2/2  and  2/3  in  (12)  are  the  remaining 
roots  of  cubic  (6).     To  give  a  formal  proof,  note  that,  by  (9)-(ll), 


40  THEORY  OF  EQUATIONS  ICh.  iv 


(13) 


2/1  +  2/2  +  2/3   =  C, 

2/i2/2  + 2/12/3+ 2/22/3  =  (a:i+X2  +  0:3  +  X4)  (0:13:22:3  +  •  •  •  +  0:20:30:4)  -  4  0:10:20:30:4 
=  6d  —  4  e, 

2/12/22/3  =  (0:10:20:3  +•••)"  +  o-iO-20-30-4K^i  +  •  •  0^  -  4  (0:10:2+  "  •  •)! 
=  d2_^e(6-  -  4  c). 


Hence  by  Ex.  7,  p.  32,  or  by  Ch.  VI,  §1,  yi,  1/2,  2/3  ^re  the  roots  of  (6). 


EXERCISES 

1.  Why  is  it  sufficient  for  the  last  proof  to  verify  merely  the  first  two  relations 
(13)? 

2.  In  Lagrange's  solution  of  quartic  (1),  we  begin  by  sho\\dng  that  the  num- 
bers (12)  are  the  roots  of  cubic  (6)  by  using  (13)  and  the  theorem  of  §  2.  Let  a 
root  iji  be  found.  Then  we  obtain  .riX2  =  Zi  and  .r3X4  —  Zj  as  the  roots  of  2^  —  yiZ 
+  e  =  0.     Next,  Xi  +  xz  and  .T3  +  Xi  are  found  from 

(xi  +  Xi)  +  (xa  +  X4)  ^  -b,     Z2{xi  +  X2)  +  Zi{x3  +  Xi)  =  -d. 

Hence  Xi  and  X2,  Xs  and  Xi  are  found  by  solving  quadratic  equations.    Give  the 
details  of  this  work. 

3.  If  the  t  corresponding  to  each  root  of  (6)  is  zero,  equation  (1)  has  all  its 
roots  equal.  For,  by  (3),  the  y's  all  equal  c  —  lb-.  By  (13),  3  ?/i  =  c,  3  yi^  = 
bd  —  4:6.  Hence  c  =  §6^,  5?  b*  =  bd  —  4:6.  Eliminating  e  between  the  latter 
and  {I  ¥Y  =  yi^  =  ¥e  —  4ce  -{-  d^,  which  follows  from  yi  =  c  —  Ib^  and  (13),  we 
get  (t\  ¥  -d)^  =  0.    Then  (1)  equals  {x  +  {  b)'  =  0. 

4.  Prove  that  Ex.  3  is  true  by  showing  that  t-  =  (xi  +  X2  —  X3  —  X4)-. 

5.  Solve  :i^  -\-  px  +  q  =  0  ip  9^  0)  by  choosing  c  so  that  the  quartic 

(x  -  c)i.r'  +  px  +  q)  =  0 

shall  have  as  its  resolvent  cubic  (6)  one  reducible  to  the  form  z^  =  constant.     Here 
(6)  is 

y^  -  py-  +  c{cp  +  ^q)y  -  c-p"^  —  2  cpq  —  q-  -{- <^q  =  0. 

To  remove  the  second  term,  set  ?/  =  z  +  p/3.    We  get 

z^  -\-  Az  -\-  (?q  —  I  C'p-  —  cpq  —  ([•  —  ^-y  p'  =  0, 

where  A  =  p&  -\-2tcq  —  \  p^.    We  are  to  make  .4=0;   thus 


§41  QUARTIC  EQUATIONS  41 

since  c^  +  cp  +  5  =  36  cR/p'^.  Our  quartic  has  the  root  c  and  hence  by  (81), 
with  b  replaced  by  —c,  also  the  root  Kc  +  0  —  c,  where  i^  =  c^  —  4  p  +  4  y. 
Hence  the  given  cubic  has  the  root 


i(i-c)  =  Vz-  ip  +  ic^-lc, 
which  may  be  reduced  to  Cardan's  form  (Atner.  Math.   Monthly,  1898,  p.  38). 

4.   Discriminants.     Replacing  y  by  F  +  c/3  in  (6),  we  get 

(14)  Y'-{-PY-\-Q  =  0, 
in  which 

(15)  P  =  6d  -  4  e  -  i  c2,     Q  =  -¥e  +  lhcd-\-^  ce-(P  -  /f  c^. 
Hence  (Ch.  Ill,  §  3), 

(1/1  -  y^YiUi  -  ysKy^  -  y.Y  =  -4^^  _  21  Q\ 
By  (12) 

Vi  -  yi  =  {xi  -  Xi){x2  -  xz), 

(16)  yi  -  tjz  =  {xi  -  a;3)(a:;2  -  Xi), 
2/2  -  2/3  =  (xi  -  a;2)(x3  -  3:4). 

The  discriminant  A  of  the  quartic  (1)  is  defined  to  be 

(17)       A  =  {xi  —  0:2)2(3:1  —  xsYixi  —  Xiy{x2  —  X3y(x2—Xiy{x3—X4y. 

It  therefore  equals  the  discriminant  of  (14) : 
(18)  A  =  -4P3_27Q2. 

Any  quartic  equation  and  its  resolvent  cubic  have  equal  discriminants. 

Some  writers  define  the  discriminant  of  (1)  to  be  A/256  and  that  of  a  cubic  to 
be  —A/27.  In  suppressing  these  numerical  factors,  we  have  spared  the  reader 
a  feat  of  memory,  simplified  the  important  relation  between  the  discriminants  of 
a  quartic  equation  and  its  resolvent  cubic,  and  moreover  secured  uniformity  with 
most  of  the  books  to  which  we  shall  have  occasion  to  refer  the  reader.  Finally, 
we  note  that  in  applications  to  the  theory  of  numbers,  the  insertion  of  the  numer- 
ical factors  is  undesirable  and  in  special  cases  unallowable  (c/.  Bull.  Arner.  Math. 
Soc,  vol.  13,  1906,  p.  1). 

EXERCISES 

1.   For  ax*  -{-  bx^  -{-  cz^  -\-  dx  -\-  e  =  0,    P  =  p/a'^,  Q  =  q/a^,  where 
p  =  bd  —  4iae  —  c^/3,     q  =  —  6^e  +  5  bed  -\-  f  ace  —  ad^  —  ^j  c*. 
The  discriminant  is  defined  to  be  a® A;  it  equals  —  4  p'  —  27  q^. 


42  THEORY  OF  EQUATIONS  ICh.  IV 

2.  If  X  and  y  are  interchanged  in 

f  =  ax*  +  hxhj  +  cx-if-  +  c^.f//^  +  ei/, 

a  function  is  obtained  which  may  also  be  derived  from  /  by  merely  interchanging 
a  with  e,  and  b  with  d.  Show  that  the  latter  interclianges  leave  p,  q  and  the  dis- 
criminant unaltered. 

3.  Since  the  sum  Fi  -f  F2  +  Fs  of  the  roots  of  a  reduced  cubic  is  zero, 

Fi  =  HFi  -  F2)  +  §  (Fi  -  F3),  .  .  .  , 

and  any  root  and  hence  any  function  of  the  roots  is  expressible  as  a  function  of  the 
differences  of  the  roots.  Thus  P  and  Q  in  (15)  are  functions  of  Fi  —  F2,  etc., 
and  hence  of  yi  —  y^,  etc.  Using  (16),  show  that  p  and  q  equal  polynomials  in 
the  differences  of  Xi,  .  .  ,  ,  X4. 

4.  When  x  is  replaced  by  x  +  ty,  let  /  of  Ex.  2  become 

/'  =  a'x"  +  h'xhj  +  •  •  •  +  e'y\ 
Show  by  Ex.  3  that  p  and  q  equal  the  corresponding  functions 

V'  =  h'd'  -  4  a'e'  -  c'V3,     q'  =  -b'¥  +  •  •  •  . 

5.  The  results  in  Exs.  2  and  4  are  special  cases  (used  in  a  short  proof)  of  a  gen- 
eral theorem :  When  x  is  replaced  by  Ix  +  my  and  y  by  rx  +  sy,  let  /  become  /'. 
Then,  using  the  notations  of  Ex.  4,  we  have  p'  =  D*^p,  q'  =  D^q,  where  D  =  Is  —mr. 
Hence  p  and  q  are  called  invariants  of  /.  Verify  the  theorem  for  the  case  when  x 
is  replaced  by  Ix,  y  by  y. 

6.  The  discriminant  is  an  invariant  and  the  factor  is  D^'^. 

7.  Using  oax*  +  4  aix^y  +  6  aixh/  +  4  asxy^  +  a^y*  in  place  of  the  former  /, 
show  that  p=— 4/,  g=16J,  where 

/  =  0004  —  4  OiOs  "h  3  02",     J  =  000204  +  2  010203  —  aoOa"  —  01-04  —  02^. 

In  (14)  set     F  =  2  z/a;    then    z^  —  Iz  +  2J  =  0.     The    discriminant    is 

256  (P-27J2). 

5.  Descartes'  Solution  of  the  Quartic  Equation.  Replacing  x  bj' 
2  —  6/4  in  the  general  quartic  (1),  we  obtain  a  reduced  quartic  equation 

(19)  s!"  -j-  qz"  +  rz  -\-  s  =  0, 

lacking  the  term  with  ^.  We  shall  prove  that  we  can  express  the  left 
member  of  (19)  as  the  product  of  two  quadratic  factors  * 

(22  -\-  2  kz  -\-  l){z''  -  2kz  +  m)  =  z'  +  (I  -{-  m  -  4:  k'-)z-  -{-  2  k{m  -  l)z  -f  Im. 

*  If  the  coefficients  of  z  be  denoted  by  k  and  —k  (jxs  is  usually  done),  the  expres- 
Bions  (23)  for  the  roots  must  be  divided  by  2.  But  the  identification  with  Euler'a 
Bolution  is  then  not  immediate. 


§61  QUARTIC  EQUATIONS  43 

The  conditions  are 

I  -\-  m  —  4:k^  =  q,     2  k  {m  —  l)  =  r,     Im  =  s. 
If  k  7^  0,  the  first  two  give 

Then  Im  =  s  gives 

(20)  64  A:«  +  32  qk^  +  4  (g^  -  4  s)k^  -  r^  =  0. 

The  latter  may  be  solved  as  a  cubic  equation  for  k"^.     Any  root  k^  9^  0 
gives  a  pair  of  quadratic  factors  of  (19) : 

(21)  22±2A;2  +  i5  +  2A;2T^. 

The  4  roots  of  these  two  quadratic  functions  are  the  4  roots  of  (19).  If 
q  =  r  =  s  =  0,  every  root  of  (20)  is  zero  and  the  discussion  is  not  valid;  but 
the  quadratic  factors  are  then  evidently  z'^,  z^. 

EXERCISES 

1.  For  2^  -  3  ^2  +  6  2  -  2  =  0,  (20)  becomes 

64  A;«  -  3-32  k^  +  4-17  k""  -  36  =  0. 

The  value  A;^  =  1  gives  the  factors  2-  +  22  —  1,  2^  —  22  +  2,  with  the  roots 
-1±V2,     1±V^. 

2.  Solve  2*  -  2  22  -  8  2  -  3  =  0. 

3.  Solve  2^  -  IO22  -  2O2  -  16  =  0. 

4.  Solve  x-*  -  8  x^  +  9  x^  +  8  X  - 10  =  0. 

6.   Symmetrical  Form  of  Descartes'  Solution.     To  obtain  this  sym- 
metrical form,  we  use  all  three  roots  A;!^,  A'2^,  k-^  of  (20).     Then 

k^  +  ki  +  ^32  =  -1 5,     k^k-^-k^"-  =  rV64. 

It  is  at  our  choice  as  to  which  square  root  of  k^  is  denoted  by  -\-k\  and 
which  by  —  fci,  and  likewise  as  to  -^.k-i,  ±ikz.  For  our  purposes  any 
choice  of  these  signs  is  suitable  provided  the  choice  give 

(22)  kxk'.kz  =  -r/8. 

Let  ki  5^  0.     The  quadratic  function  (21)  is  zero  for  k  =  ki  if 


44  THEORY  OF  EQUATIONS  (Ch.  iv 

Hence  the  four  roots  of  the  quartic  equation  (19)  are 

(23)  /vi  +  ^-2  +  A-3,     ki-k2-kz,     -/u  +  A-2- A-3,     -k.-ko  +  kz. 
Writing  ¥  =  y,  we  see  that,  if  yi,  y-i,  y^  arc  the  roots  of 

(24)  Uy-"  +  32g!y2  +  4  (r/  -  4s)?/  -  r^  =  0, 
then  the  roots  of  (19)  are  the  four  values 

(25)  z  =  V^i  +  Vy<>  +  V?73, 

obtained  by  using  all  of  the  combinations  of  the  square  roots  for  which, 
by  (22), 

(26)  V^  VJ^>  VFs  =  -  V8. 

We  have  deduced  Euler's  solution  (Ex.  1)  from  Descartes'. 

EXERCISES 

1.  Assume  with  Euler  that  quartic  (19)  has  a  root  of  the  form  (25).  Square 
(25),  transpose  the  terms  free  of  radicals,  square  again,  and  show  that 

2«  -  2  (i/i  +  2/2  +  yz)  2-  -  S  2  Vyi  Vy.  Vys  +  (^/i  +  ^2  +  I/s)" 

-  4  (i/i7/2  +  yiys  +  yiyd  =  0. 

From  the  relations  obtained  by  identifj'ing  this  with  (19),  show  that  yi,  y^,  ys  are 
the  roots  of  the  cubic  (24)  and  that  (26)  holds. 

2.  Solve  Exs.  1-4  of  the  preceding  set  by  use  of  (23). 

3.  In  the  theory  of  inflexion  points  of  a  plane  cubic  cuiwe  occurs  the  quartic 
equation  z*  —  Sz-  —  t  Tz  —  i\  S-  =  0.     Show  that  (24)  now  becomes 

(-^■-'   -(0-(f)' 

and  that  the  roots  of  the  quartic  are 

where  the  signs  are  to  be  chosen  so  that  the  product  of  the  three  summands  equals 
+  T/Q.     Here  lo  is  an  imaginary  cube  root  of  miity. 

4.  The  discriminant  A  of  the  quartic  equation  (19)  equals  the  quotient  of  the 
discriminant  D  of  (24)  by  4^  For,  the  six  differences  of  the  roots  (23)  are 
2  {ki  ±  A-2),  2  (A-i  ±  /:3),  2  (^-2  ±  k,).     Thus  A  -  4«  L,  where 

L  =  (A;,2  -  ki^Yiki''  -  ki^Yiki-  -  kz^y  =  {y,  -  y^y-{yi  -  yzfiy^  -  yz)\ 

By  definition,  D  =  64^1.     Hence  D  =  4«  a. 


§7]  QUARTIC  EQUATIONS  45 

5.  Give  a  second  proof  of  Ex.  4  by  setting  y  =  2/4  in  (24)  and  then  z  =Y  —  2  q/3. 
We  obtain  (14),  in  which  now  b  =  0,  c  =  q,  d  =  r,  e  =  s.  The  discriminant  of 
(14)  equals  A.     Hence  A  =  (^i  —  Z2)-  •  •  •   =  4^L  =  Z)/4«. 

6.  If  a  quartic  equation  has  two  pairs  of  conjugate  imaginary  roots,  its  dis- 
criminant A  is  positive.     Hence,  if  A  <  0,  there  are  exactly  two  real  roots. 

7.  Theorem.*  A  quartic  equation  (19)  with  q,  r,  s,  real,  r  ^  0,  and 
with  the  discriminant  A,  has 

4  distinct  real  roots  if  q  and  4  s  —  5^  are  negative  and  A  >  0, 

no  real  root  if  q  and  4  s  —  5^  are  not  both  negative  and  A  >  0, 

2  distinct  real  and  2  imaginary  roots  if  A  <  0, 

at  least  2  equal  real  roots  ?/  A  =  0. 

Since  the  constant  term  of  the  cubic  equation  (24)  is  negative,  at  least 
one  of  its  roots  is  a  positive  real  number.  Let,  therefore,  yi  >  0,  so  that 
2/22/3  >  0.     Thus  ki  =  Vyi  is  real.     There  are  four  possible  cases  to  consider. 

(a)  7/2  and  yz  positive.  Then  each  kj  =  V^/yis  real  and  the  roots  (23) 
of  the  quartic  equation  are  all  real. 

(p)  y-i  =  yz  <  0.  Then  A-o  =  ±^-3  is  a  pure  imaginary.  If  k2  =  kz, 
the  first  two  roots  (23)  are  imaginary  and  the  last  two  are  real  and  equal. 
If  A'2  =  —  kz,  the  reverse  is  true. 

(c)  2/2  and  2/3  distinct  and  negative.     The  roots  (23)  are  all  imaginary. 

{d)  2/2  and  yz  conjugate  imaginaries.  Then  ki  is  imaginary  and  conju- 
gate with  either  A'3  or  —  As,  so  that  one  of  the  numbers  A'2  +  A3  and  A2  —  A:3  is 
real  and  the  other  imaginary.     Just  two  of  the  roots  (23)  are  real. 

Now,  if  A  =  0,  at  least  two  ?/'s  are  equal  by  Ex.  4  of  the  last  set.  Thus 
we  have  case  (h)  or  a  special  case  of  (a).  In  either  case,  the  quartic  has 
at  least  two  equal  roots,  by  (17),  and  they  are  real  in  both  cases. 

Henceforth,  let  A  ?^  0.  By  the  same  Ex.  4,  A  has  the  same  sign  as 
the  discriminant  D  of  the  cubic  equation  (24).  If  A  <  0,  we  have  case 
{d).  Finally,  let  A  >  0,  so  that  yi,  yo,  ys  are  real.  If  q  is  negative  and 
q^  —  4:  s  is  positive,  equation  (24)  has  alternately  positive  and  negative 
coefficients  and  hence  has  no  negative  root,  so  that  we  have  case  (a). 
But  if  q  and  4  s  —  g-  are  not  both  negative,  the  coefficients  are  not  alter- 
nately positive  and  negative,  so  that  the  roots  yi,  y-i,  ys  are  not  all  posi- 
tive,** and  we  have  case  (c). 

*  Proved  by  Lagrange  by  use  of  the  equation  whose  six  roots  are  the  squares  of  the 
differences  of  the  roots  of  (19),  Resolution  des  equations  numeriques,  3d  ed.,  p.  42. 
**  The  coeflacients  are  —  (//i  +  2/2  +  ys),  yiU'i  +  2/i2/3  +  UzUi,  —  yiyiUz- 


46  THEORY  OF  EQUATIONS  [Ch.  iv 

EXERCISES 

1.  Apply  this  theorem  to  the  quartic  equations  in  Exs.  1-4,  p.  43. 

2.  Verify  that  a  quartic  equation  (19)  with  two  pairs  of  equal  imaginary  roots 
has  r  =•  0.    Deduce  the  last  case  of  the  theorem. 

3.  Why  does  the  theorem  imply  its  converse? 


t  CHAPTER  V 
The  Fundamental  Theorem  of  Algebra 

l.f   Theorem.     Evenj  equation  with  complex  coefficients 

(1)  Kz)  =  2»  +  a,2«-i  +  .  .  .  +  a„  =  0 

has  a  complex  {real  or  imaginary)  root. 

For  n  =  2,  3,  or  4,  we  have  proved  this  theorem  by  actually  solving 
the  equation.  But  for  n  =  5,  the  equation  cannot  in  general  be  solved 
algebraically,  i.e.,  in  terms  of  radicals. 

We  shall  first  treat  the  case  in  which  all  of  the  coefficients  are  real. 
Relying  upon  geometrical  intuition,  we  have  seen  in  Exs.  3,  5,  p.  14, 
that  there  is  a  real  root  if  n  is  odd,  or  if  both  n  is  even  and  a„  is  negative. 
But,  as  in  the  cases  of  certain  quadratic  equations  and  s'*  +  2^  +  5  =  0, 
an  equation  of  even  degree  may  have  no  real  root.  No  proof  of  the 
theorem  for  all  cases  has  been  made  by  such  elementary  methods. 

The  proof  here  given  of  the  theorem  that  any  equation  with  real  co- 
efficients has  a  complex  root  is  essentially  the  first  proof  by  Gauss  (1799 
and  simphfied  by  him  in  1849). 

We  are  to  prove  that  there  exists  a  complex  number  z  =  x  -\-  yi  such 
that  f(z)  =  0.     We  may  write 

(2)  Kz)  =  X  +  Yi, 

where  X  and  Y  are  polynomials  in  x  and  y  with  real  coefficients.     We 
are  to  show  that  there  exist  real  numbers  x  and  y  such  that 

(3)  X  =  0,     Y  =  0. 

For  example,  if  fiz)  =  2"  -  4  s-''  +  9  2'  -  16  2:  +  20,  then 

X  =  x"  -  6  xY  +  ?/  -  4  a;3  +  12  xif  +  9  x^  -  9  ?/  -  16  a;  +  20, 
hY  =  2xhj  -2  X7f  -  Qxhj  +  2  ?/^  +  9  x?/  -  81/. 

The  graph  of  F  =  0  is  the  x-axis  {y  =  0)  and  the  graph  (indicated  by  the  dotted 
curve  in  Fig.  16,  asymptotic  to  the  lines  x  =  1  and  ?/  =  ±  x)  of 

2  (x  ^  1)7/ =  2x3  -  6x2  +  9x  -  8. 
47 


48 


THEORY  OF  EQUATIONS 


[Ch.  V 


Note  that  there  is  no  real  y  for  x  between  1  and  1.73.     Since  X  =  0  is  a  quadratic 
equation  in  y"^,  its  graph  is  readily  drawn.    There  is  no  real  y  ior  x  =  0.05  and  1.6 


Fig.  16 


and  the  intermediate  values.     Cases  in  wliicli  the  values  of  y-  are  positive  and 
rational  are 


X 

-4 

-  2 

-  1 

0 

2 

3 

y2 

5,  148 

2.5,  54.5 

2,25 

4,5 

1,8 

1,26 

The  graphs  cross  at  the  points  (0,  2),  (0,  —2),  (2,  1),  (2,  —1),  and  the  roots  of 
f{z)  =  0  are  z  =  ±2  t,  2  ±  {. 

We  shall  employ  also  the  trigonometric  f onn  of  z : 

(4)  z  =  r(cos  6  -\-  i  sin  d), 


5  11  FUNDAMENTAL   THEOREM  OF  ALGEBRA  49 

where  0  ^6  <2t.     Set  t  ^  tan  ^  6.     Then 

2t     _  2tan^ 
rT^~   sec2 1  ( 

2f 


tan  0 


Thus 


-  =  2  sin  ^  ^  •  COS 

^0  =  sin 

cos  9 

sin  9 
tan  6 

1  +  ^2 

1  -f' 

r  (1  +  tiy 


z  = 


1  -{-f 
Hence  by  (1)  and  (2), 

(H-/-)"(X+yO=r"(l+^0'"+«i^"~Hl  +  ^0'"~'(l+^^)+  •  •  •  +  an  (1+^2)". 
Expanding  the  terms  on  the  right  by  the  binomial  theorem,  we  get 

(5)  X  =  ,M^,     Y=      ^(^) 


Avhere  F(t)  is  a  polynomial  in  t  of  degree  2  n,  and  (7(0  a  polynomial  in  t 
of  degree  less  than  2  /?,  each  with  coefficients  involving  r  integrally. 

Each  point  {x,  y),  representing  (Ch,  II,  §  8)  a  complex  number 
z  —  X  -\-  iji  having  the  modulus  r,  lies  on  the  circle  x"^  -{-  y^  =  r^  with  radius 
r  and  center  at  the  origin  of  the  rectangular  coordinate  system.  To  find 
the  points  on  this  circle  for  which  A"  =  0  or  F  =  0,  we  solve  F{t)  =  0  or 
G{t)  =  0  (in  which  r  is  now  a  constant),  and  note  that  to  each  real  root  t 
corresponds  a  single  real  value  of  sin  6  and  a  single  real  value  of  cos  6, 
consistent  with  that  of  sin  Q,  and  hence  a  single  point  {x  =  r  cos  d, 
y  =  r  sin  0) .  But  an  equation  of  degree  2  n  has  at  most  2  n  distinct 
roots  (Ch.  I,  §  15).  Since  the  degree  of  G{t)  is  less  than  that  of  the  de- 
nominator of  Y  in  (5),  the  root  ^  =oo  of  F  =  0  must  be  considered  in 
addition  to  the  roots  of  G{t)  =  0  already  examined;  for  t  =cc,  6  =  ir  and 
the  point  is  (  — r,  0).  Thus  neither  X  nor  F  is  zero  for  more  than  2n 
points  of  the  circle  with  center  at  the  origin  and  a  given  radius  r.  By 
proper  choice  of  r,  this  circle  will  have  an  arc  lying  within  any  given 
region  of  the  plane.  Hence  neither  X  Jior  Y  is  zero  at  all  points  of  a  region 
of  the  plane. 

From  (4)  and  DeMoivre's  Theorem  (Ch.  II,  §  10),  we  have 

^k    —   yk  (^(jQg  ^.^  _J_  I  gjjj  ]^QY 

Hence,  by  (1)  and  (2), 

F=r"  sin  n^  + air  "-1  sin  (n— 1)0  + aor"-^  sin  (71  —  2)0+  •  •  •  +a„_irsin0. 


50  THEORY  OF  EQUATIONS 

Let  g  be  the  greatest  of  the  numerical  values  of  aj,  .  . 
1  D  I  denotes  the  numerical  value  of  the  real  number  D 

y=r»(sinn0  +  Z)),     \D\^g(^l  +  ^,+  •  • 


provided  r  >  1.  If  c  is  a  positive  constant  <  1  and  if  r  >  1  +  g/c, 
then  I  D  I  <  c.  Hence  for  all  angles  6  for  which  sin  nd  is  numerically  greater 
than  c,  Y  has  the  same  sign  as  its  first  term  ?"'  sin  7i6  when  r  exceeds  the 
constant  1  +  g/c. 

In  our  example,  we  have 

Y  =  ?•■»  sin  4  9  —  4  r^  sin  3  0  +  9  r-  sin  2  0  —  16  r  sin  e. 

The  limit  1  +  16/c  for  r  exceeds  17  and  is  larger  than  is  convenient  for  a  drav/ing. 
But  for  r  S  10, 

4       0       16 

Y  =  r*{sin4d  +  D),     \D\= -  +  -+  —  =  0.4  +  0.09  +  0.016. 

sin  30°  24',  let  C  be  the  number  of  radians  in  7°  36'. 
Thus  c  =  sin  4  C.  The  positive  angles  d  (d  <  2  tt) 
for  which  sin  4  d  exceeds  sin  4  C  numerically  are 
those  between  C  and  |  tt  —  C,  between  J  tt  +  C 
and  h  TT  —  C,  between  ^  ir  -{-  C  and  f  tt  —  C,  ,  .  .  , 
between  iir  -\-  C  and  2  w  —  C.  For  any  such 
angle  6  and  for  r  =  10,  F  has  the  same  sign  as 
sin  4  e  and  hence  is  alternately  positive  and 
negative  in  these  successive  intervals,  the  solid 
\+  _/      arcs  in  Fig.  17.     Denote  by  0,  1,  2,  .  .  .  ,  7  the 

\  /        points  on  the  circle  with  center  at  the  origin  and 

••  \  /^  TT       ^  TT  4    TT 

■^  ""^v.--  4.^"  '  radius  10  whose  angles  d  are  0,-,  — -,  .  .  .  ,  — ?-» 

^- — ,S>^  4      4  4 

^  respectively. 

Fig.  17 

In  the  general  case,  denote  by  0,  1,  2,  ...  , 

2  n  —  1  the  points  with  the  angles 

TT   2  X  (2  n  -  1)  TT 

u,     ,       ,  .  .  .  , 

n     n  n 

on  the  circle  with  center  at  the  origin  and  radius  a  constant  r  exceeding 
the  above  value  1  +  g/c.  Let  nC  be  the  positive  angle  <  7r/2  for  which 
sin  nC  =  c.     We  define  the  neighborhood  of  our  A;th  point  of  division  on 


Taking  c  = 

0.506 

=  sin  30°  2 

/ 

/ 
/ 

1- 

1 

A 

4I 
1 

1 

10 

§  n 


FUNDAMENTAL   THEOREM  OF  ALGEBRA 


51 


the  circle  to  be  the  arc  bounded  by  the  points  whose  angles  are  kTr/n  —  C 
and  kir/n  +  C.  In  Fig.  17  for  our  example  with  ri  =  4,  each  neighborhood 
is  indicated  by  a  dotted  arc.  In  the  successive  arcs  (marked  by  solid 
arcs)  between  the  neighborhoods,  Y  is  alternately  positive  and  negative, 
since  it  has  in  each  the  same  sign  as  sin  nd. 

It  is  easily  seen  that  sin  ^,  sin  2  ^,  .  .  .  ,  sin  nd  are  continuous  functions 
of  6  (a  fact  presupposed  in  interpolating  between  values  read  from  a 
table  of  sines).  Since  r  is  now  a  constant,  Y  is  therefore  a  continuous 
function  of  6,  and  has  a  single  value  for  each  value  of  d.  But  Y  has  oppo- 
site signs  at  the  two  ends  of  the  neighborhood  of  any  one  of  our  points  of 
division  on  the  circle.  Hence  (as  in  Ch.  I,  §  12),  Y  is  zero  for  some  point 
\vithin  each  neighborhood,  and  at  just  one  such  point,  since  Y  was  shown 
to  vanish  at  not  more  than  2  n  points  of  a  circle  with  center  at  the  origin. 
We  shall  denote  the  points  on  the  circle  at  which  Y  is  zero  by 


^0,  ^1, 


P2n- 


For  our  example,  these  points  Po,  .  .  .  ,  Pi  are  given  in  Fig.  18,  which  shows 
more  of  the  graph  of  F  =  0  than  was  given  in  Fig.  16,  but  now  shows  it  with  the 


Fig.  18 


scale  of  length  reduced  in  the  ratio  4  to  1  (to  have  a  convenient  circle  of  radius  10). 
We  have  shaded  the  regions  in  which,  as  next  proved,  Y  is  positive. 


62  THEORY  OF  EQUATIONS  [Ch.  v 

Let  the  constant  r  be  chosen  so  large  that  X  also  has  the  same  sign  as 
its  first  term  r"  cos  nd,  for  6  not  too  near  one  of  the  values  7r/(2  n),  3  tt/  (2  n), 
5ir/{2n),  .  .  .  ,  for  which  cos  w^  =  0.  Since  these  values  correspond  to 
the  middle  points  of  the  arcs  (01),  (12),  .  .  , ,  no  one  of  them  lies  in  a 
neighborhood  of  a  division  point  0,  1,  ...  .  Now  cos  n0  =  +1  or  —  1 
when  6  is  an  even  or  an  odd  multiple  of  ir/7i,  respectively.  Hence  X  is 
positive  in  the  neighborhood  of  the  division  points  0,  2,  4,  .  ,  .  ,  2  n  —  2 
and  thus  at  Po,  -P2,  P4,  •  •  • ,  but  negative  in  that  of  1,  3,  5,  .  .  .  ,  2  n  —  1 
and  thus  at  Pi,  P3,  P5,  •  •  •  • 

We  saw  that  Y  is  not  zero  throughout  a  region  of  the  plane.  Hence  there 
is  a  region  in  which  Y  is  everj^vhere  positive  (called  a  positive  region), 
and  perhaps  regions  in  which  Y  is  everywhere  negative  (called  negative 
regions),  while  Y  is  zero  on  the  boundary  lines. 

In  Fig.  18  for  our  example,  there  are  three  positive  (shaded)  regions,  the  two 
with  a  single  point  in  common  being  considered  distinct,  and  three  negative  (un- 
shaded) regions.  Consider  that  part  of  the  boundary  of  ^2^30  which  lies  inside 
the  circle.  At  every  point  of  it,  Y  is  zero.  Now  X  is  negative  at  Pz  and  positive 
at  P2  and  hence  is  zero  at  some  intermediate  point  a  on  this  boundary.  Hence 
at  a  both  X  and  Y  are  zero,  so  that  a  represents  a  complex  root  (in  fact,  2  i)  of 
m  =  0. 

To  extend  the  last  argument  to  the  general  case,  let  R  be  the  part  in- 
side our  circle  of  a  positive  region  having  the  points  P2  h  and  P2  h+i  on  its 
boundary.  The  points  of  arc  Po  hPi  h+\  may  be  the  only  boundary  points 
of  R  lying  on  the  circle  (as  for  P2P3a  and  P^Pid  in  Fig.  18),  or  else  its 
boundary  includes  at  least  another  such  arc  P2  ;tP2  k+i  (as  shaded  region 
PiP^bPePTC  in  Fig.  18).  In  the  first  case,  X  and  Y  are  both  zero  at  some 
point  (a  or  d)  on  the  inner  boundary,  since  X  is  negative  at  P2A+1  and 
positive  at  P2  h  and  hence  zero  at  an  intermediate  point.  In  the  second 
case,  a  point  moving  from  P2  h  to  P2  h+i  along  the  smaller  included  arc  and 
then  along  the  inner  boundary  of  R  until  it  first  returns  to  the  circle 
arrives  at  a  point  P2A;  of  even  subscript  (as  in  the  case  of  P4P56P6).  In- 
deed, if  a  person  travels  as  did  the  point,  he  aaiII  always  have  the  region 
R  at  his  left  and  hence  will  pass  from  Po  k  to  Po  k+i  and  not  vice  versa.  Since 
X  is  negative  at  P2  /,+i  and  positive  at  P2  k,  it  (as  also  F)  is  zero  at  some 
point  b  on  the  part  of  the  inner  boundary  of  R  joining  these  two  points. 
Hence  b  represents  a  root  of  f{z)  =  0.  Thus  in  either  of  the  two  pos- 
sible cases,  the  equation  has  a  root,  real  or  imaginary. 


§21 


FUNDAMENTAL   THEOREM  OF  ALGEBRA 


53 


2.t  It  remains  to  prove  that  an  equation  F{z)  =  0,  not  all  of  whose  co- 
efficients are  real,  has  a  complex  root.  By  separating  each  imaginary  coeffi- 
cient into  its  real  and  purely  imaginary  parts,  we  have  F(z)  =  P  -{-  Qi, 
where  P  and  Q  are  polynomials  in  z  with  real  coefficients.  Let  G{z)  =  P  —  Qi. 
The  equation 

F{z)  .  G{z)  ^P^-\-Q^  =  0 

has  real  coefficients  and  hence  has  a  complex  root  z  =  a  -\-  hi.  If  this  is  a 
root  of  F(z)  =  0,  our  theorem  is  proved.  If  it  is  not,  then  G(a  +  bi)  =  0. 
Then  by  Ch.  II,  §  16,  F(a  —  hi)  =  0,  and  the  given  equation  has  the  root 
a  —  hi. 

EXERCISES 

l.f  For  z^  =  11  -\-  2{,  draw  the  graphs  of  A'  =  0,  F  =  0  and  locate  the  three 
roots  of  the  cubic  equation  in  z. 


Fig.  19 


2.t  For  z^  —  4i  z  —  2  =  0,  Y  =  r^  sin  5  e  —  4  r  sin  6.  Using  polar  coordinates, 
show  that  the  graph  of  F  =  0  gives  the  boundaries  of  the  regions  in  Fig.  19:  first 
plot  the  horizontal  line  corresponding  to  sin  6  =  0,  and  then,  using  various  angles 
e  {e  i?^  0,  tt),  find  by  logarithms  tlK  corresponding  positive  r  from 

4  sin  6 


r*  = 


sin  5  d 


54  THEORY  OF  EQUATIONS  |Ch.  v 

To  fiiKl  the  points  on  these  boundaries  (F  =  0)  for  whicli  also 

X  =r^cos5d  —  4ircosd  —  2  =  0, 

replace  r*  by  the  earlier  expression.     We  get 

^/•„        r  ^                -rxo-r                      sin  59 
4  ?-(sin  d  cos  5  d  —  cos  d  sm  5  9)  =  2  sni  5  0,     r  = ^ • 

2  SHI  4:  0 

Comparing  the  fourth  power  of  this  fraction  with  that  for  r*,  we  get 

sin^  5  0  =  64  sin  9  sin^  4  0, 

which  holds  for  6  =  85°  21'  30"  or  its  negative.     We  then  get  r  and  therefore 
the  roots 

€,5  =  0.11679  ±  1.43851. 

On  the  horizontal  line  are  three  real  roots,  best  found  by  methods  of  approxima- 
tion given  later: 

a  =  1.518512,     0  =  -0.5084994,     7  =  -1.2435964. 

(H.  Weber  and  J.  Wellstein,  EncyJdopddie  der  Elementar-Mathematik,  ed.  1,  I, 
p.  212,  p.  296.) 

3.t  Other  References.  For  proofs  of  the  fundamental  theorem  by  Gauss, 
Cauchy  and  Gordan,  see  Netto,  Vorlesungen  fiber  Algebra,  I,  p.  25,  p.  173.  The 
shortest  proofs  are  by  the  use  of  the  theory  of  functions  of  a  complex  variable, 
and  may  be  found  in  texts  on  that  subject.  For  an  algebraic  proof  resting  upon 
the  theory  of  functions  of  a  real  variable,  see  Weber,  Lehrbuch  der  Algebra,  2d  ed., 
vol.  1,  pp.  119-142.  See  also  Monographs  on  Topics  of  Modern  Mathematics,  1911, 
p.  201,  edited  by  Young  (article  by  Huntington).  In  the  Amer.  Math.  Monthly, 
vol.  10  (1903),  p.  159,  Moritz  has  pointed  out  hidden  assumptions  in  various  in- 
complete proofs. 


CHAPTER  VI 

Elementary  Theorems  on  the  Roots  of  an  Equation 

1.  Relations  between  the  Roots  and  the  Coefl&cients.  Given  an 
equation  in  x  of  degree  n,  we  can  divide  its  members  by  the  coefficient  of 
a:"  and  obtain  an  equation  of  the  form 

(1)  /(.r)  =x^-{-  pia:"-i  _[_  p^_^n-i  +  .  .  .  ^  p^  =  0^ 

By  the  fundamental  theorem  of  algebra  (Ch.  V),  it  has  a  root  ai,  and 
its  quotient  hy  x  —  ai  has  a  root  ao,  etc.     Thus 

(2)  Kx)  ^(x-  a,){x  -  ao)  .  .  .   (.r  -  a,), 

identically  in  x.  Since  the  polynomial  has  n  linear  factors,  each  having 
one  root,  we  shall  say  that  the  equation  has  n  roots.  These  may  not  all 
be  distinct;  exactly  w  of  them  equal  ai,  if  ai  is  a  root  of  multiplicity  m, 
i.e.,  if  exactly  m  of  the  linear  factors  in  (2)  equal  x  —  oti.     Next, 

{x  —  ai)(x  —  ai)  ^  X-  —  (ai  +  a2)x  +  ai«2, 

{x  —  ai)  (x  —  0:2)  (x  —  as)  =x^  —  (ai  +  0:2  +  az)x'^  +  (aia-y  +  «ia3  +  ccza^jx  —  aia2(Xz. 

Thus  for  n  =  2  or  3,  we  see  that  the  product  (2)  equals 

(3)  re"-  (ai  +   •   •  •    +  a;„).-C"-i  +  (aia2  +  Uia^  +  aoaa  +    •   •  •    +  an-ian)x"-' 

—  {aia2a3  +  aia2ai-\-  •  •  •  -\-an-2an-ian)x"~^-{-  •  •  •  +(  — l)"aia2  -  '  •  a„. 

Multiplying  this  by  x  —  an+i,  we  readily  verify  that  the  product  is  a 
function  which  may  be  derived  from  (3)  by  changing  n  into  n  +  1.  It 
therefore  follows  by  mathematical  induction  that  (2)  and  (3)  are  identical. 
Hence  (1)  and  (3)  are  identical,  so  that 

«1  +  ^2  +     •    •    •     +  «„   =    —  Pi, 

ai(X2  4-  0:10:3  +   •   •   •    +  a;n-io:,i  =  P2, 

(4)  0:10:20:3  +  o:iavQ;4  -f    •  •   •    4-a:„_2Q:„_iQ:„  =  —ps, 


aia2  •  •  •  q;„-iq;„  =  (— 1)"2}„. 

For  n  =  3  and  n  —  4,  the  complete  formulae  were  given  and  proved 
other\vise  in  Ex.  7,  p.  32  and  Ch.  IV,  §  2. 


55 


56  THEORY  OF  EQUATIONS  ICh.  VI 

In  an  equation  in  x  of  degree  n,  inwhicJLtJ]£^^^cicntofjt^Lisjunity,  the 
sum  of  the  roots  equals  the  negative  of  the  coefficient  of  x"~'^,  the  sum  of  the 
products  of  the  roots  two  at  a  time  equals  the  coefficient  of  x'^~'^,  the  smyi  of  the 
■products  of  the  roots  three  at  a  time  equals  the  negative  of  the  coefficieyit  of 
x"~^,  etc.;  flnaUy,  the  product  of  the  roots  equals  the  constant  term  or  its  nega^ 
live  according  as  n  is  even  or  odd. 

For  example,  in  a  cubic  equation  having  the  roots  2,  2,  5,  the  coeffi- 
cient of  X  equals  2- 2  +  2- 5  +  2- 5  =  24. 

Given  an  equation  a^x'^  +  aio:""^  +  •  •  •  =0,  Ave  first  divide  by  ao  and 
then  apply  the  theorem  to  the  resulting  equation.  Thus  the  sum  of  the 
roots  equals  —  ai/ao. 

EXERCISES 

1.  Find  the  quartic  equation  having  2  and  —2  as  double  roots. 

2.  Find  the  remaining  root  in  Exs.  1,  3,  p.  9.  

3.  If  a  real  cubic  equation  x^  —  6  .c-  +  •  •  •  =  0  has  the  root  1  +  "^—5, 
what  are  the  remaining  roots? 

4.  Form  by  the  theorem  the  equations  in  Exs.  3,  4,  p.  15. 

5.  Given  that  .r"  -  2  x^  -  5  .r^  -  6  .r  +  2  =  0  has  the  root  2  -  V3,  find 
another  root  and,  by  using  the  sum  and  product  of  the  four  roots,  form  the  quad- 
ratic equation  for  the  remaining  two  roots  (avoid  division). 

6.  Find,  by  use  of  (4),  the  roots  of  .r^  -  6  .^;3  +  13  x"  -  12  .c  +  4  =  0,  given 
that  it  has  two  double  roots. 

7.  Solve  x^  —  3  X-  —  13  .r  +  15  =  0,  with  roots  in  arithmetical  progression. 

8.  Solve  4  x^  —  16  .c-  —  9  .c  +  36  =  0,  one  root  being  the  negative  of  another. 

9.  Solve  x^  —  9  .c-  +  23  x  —  15  =  0,  one  root  being  triple  another. 

10.  Solve  .r^  —  14  x-  —  84  .t  +  216  =  0,  with  roots  in  geometrical  progression. 

11.  Solve  x*  —  2  .r^  —  21  x-  +  22  x  +  40  =  0,  with  roots  in  arithmetical  pro- 
gression.    Denote  them  by  c  —  3  6,  c  —  6,  c  +  &,  c  +  3  6. 

12.  Solve  .r*  -  6  x^  +  12  x^  -  10  x  +  3  =  0,  with  a  triple  root. 

13.  Find  a  necessary  and  sufficient  condition  that 

/(x)  =  x^  +  ^hx-  +  IHX  +  Pa  =  0 
shall  have  one  root  the  negative  of  another.    Note  that 

(a2  +  «3)  (ai  +  "3)  (ai  +  "2) 

is  obtained  by  substituting  x  =  —  pi  in  (2). 

14.  If  for  n  =  4  the  roots  of  (1)  satisfy  the  relation  aiao  =  azai,  then  pi^Pi  =  ps'^. 
Note  that  (4)  gives 

—  Pa  =  a\a2{az  +  aj)  +  aiai{ai  +  a^)  =   —  piaiao. 

15.  What  is  the  coefficient  of  y"~^  in  the  equation  ?/"+•••  =0  whose  roots 
are  ai  —  /i,  •  •  -  ,  a„  —  h,  when  the  as  are  the  roots  of  (1)?     For  what  value  of 


§2,31  THEOREMS  ON  ROOTS  OF  EQUATIONS  57 

h  is  this  coefficient  zero?  Hence  to  remove  the  second  term  of  an  equation  by 
replacing  x  by  y  -\-  h,  what  value  of  h  must  we  take?  Check  by  the  binomial 
theorem. 

16.  Find  the  equation  whose  roots  are  the  roots  of  .r^  —  6  x^  +  4  =  0  each 
diminished  by  3.     Remove  the  second  term  by  transformation. 

17.  Prove  the  binomial  theorem  by  taking  the  as  all  equal  in  (2)  and  (3)  and 
counting  the  number  of  terms  in  each  coefficient  of  (3). 

18.  Using  (1)  and  (2),  show  that 

(1  -  «i2)(l  -  a,')  •  •  •  (1  -  an')  =  il+P2  +  Pi+  •  --y-  (pi  +  2^3  +  P5  +  •  •  •)^ 

(1  +  ai2)(l  +  ao=)  •••(!  +  a,r)  =  (1  -  p,  +  p, )2  +  (p,  _  p,  +  p,-    ...  )2. 

19.  Since  .ri,  .  .  .  ,  X4,  determined  by  relations  (8)  of  Ch.  IV,  give  the  correct 
values  of  the  sums  (9)-(ll),  they  are  the  roots  of  the  quartic  equation.  Why  does 
this  give  a  new  solution  of  the  quartic? 

20.  Using  Ex.  6,  p.  32,  make  a  similar  argument  for  the  cubic. 

2.  Upper  Limit  to  the  Positive  Roots.     For  an  equation 

fix)  =  aox"  +  ai.x"-i  4-  .  .  .  +  a„=  0  (ao  5^  0) 

with  real  coefficients,  we  shall  prove  the 

Theorem.  If  ao,  ai,  .  .  .  ,  a^-i  are  each^O,  while  Ok  <  0,  and  if  G  is 
the  greatest  of  the  numerical  values  of  the  negative  coefficients,  each  real  root  is 
less  than  1  +  ^G/oq. 

For  positive  values  of  x,  f{x)  is  numerically  greater  than  or  equal  to 

aox""  -  Gix"-''  +  a;«-'^-i  +  •  •  •  +  x  +  1) 
„       „  /a;"-^+^-l\  _  a;»-^-+Mao(a;^  -  x''-')  -Gl  +G 

—  aoX  (t  I  -        1  —  ; . 

\     x  —  I    /  X  —  I 

But,  if  X  >  1,     x^  -  x^-i  =  (x  -  1)K     Hence  if  a;=l  +  VgJo'o, 
OoCa;^  -  x'^-')  ^  G,    f(x)  ^  0. 

3.  Another  Upper  Limit  to  the  Roots.  If  the  numerical  value  of  each 
negative  coefficient  he  divided  by  the  sum  of  all  of  the  positive  coefficients 
which  precede  it,  the  greatest  quotient  so  obtained  when  increased  by  unity 
gives  an  upper  limit  to  the  positive  roots  of  the  equation. 

If  the  coefficient  of  x'^  is  positive,  we  replace  x""  by 

(x  -  l)(a;'"-i  +  a;'«-2  +  .  .  .  +  a;  +  1)  +  I. 


58  THEORY  OF  EQUATIONS  [Ch.  VI 

The  argument  will  be  clearer  if  applied  to  a  partif^ular  case: 

f{x)  =  jhx''  -  pix^  +  ]hx^  +  psx-  -  PiX  +  P'o  =  0, 
where  each  pi  is  positive.     Then/(.c)  is  the  sum  of  the  terms 

Po{x  —  l)x*  +  Pq{x  —  l)x*  +  po(x  —  l)x-  4-  po(x  —  l)x  +  po{x  -  1)  +  po 
—  piX*  p2(x  —  l)x-  +  2>2(a:  —  l)a;  +  P'zix  —  1)  +  P2 

7)3(3:  -  l).r  +  Psix  -  1)  +  ps 
—  PiX  Pa. 

The  sum  of  the  terms  in  each  column  will  be  positive,  if  a:  >  1  and 

PO  {X   -    1)    -   Pl>  0,        {po  -\-p2  +  PZ){X  -   1)    -   P4   >   0, 

since  only  in  the  first  and  fourth  columns  is  there  a  negative  part.     These 
inequalities  both  hold  if 


Po  7>0  +  P2  +  P3 

EXERCISES 
Apply  the  methods  of  both  §  2  and  §  3  to  find  an  upper  limit  u  to  the  roots  of 

1.  4x^  -  8x4  +  22x5  ^  98_j.2  _  73^  +  5  =  0.  By  §2,  w  =  1  +  73/4.  By  §  3, 
u=  3,  since  1+8/4  =  3,     1  +  73/124  <  3.  _ 

2.  x^  +  4a;'»-7x2-40.r  +  l  =  0.     By  §  2,  w  =  1  +  ^^40  =  4.42.   By§3,M  =  9. 

3.  x4-5a;3  +  7x2 -8x  + 1  =  0. 

4.  x^  +  3  xs  -  4  x5  +  5  X*  -  6  x3  -  7  .x2  -  8  =  0. 

5.  x7  +  2x5  +  4x4-8.t2-32  =  0. 

6.  If  A  is  the  greatest  of  the  numerical  values  of  Oi,  .  .  .  ,  a„,  each  root  is 
less  than  1  +  A/ao.     In  the  proof  in  §  2,  set  ^-  =  1  and  replace  G  by  .4. 

7.  A  lower  limit  to  the  negative  roots  of  /(x)  =  0  may  be  found  by  appljnng 
the  above  theorems  to/(— x)  =  0.  To  obtain  a  lower  limit  to  the  positive  roots 
consider /(I /x)  =  0. 

8.  Find  a  lower  limit  to  the  negative  roots  in  Exs.  3,  4. 

9.  Find  a  lower  limit  to  the  positive  roots  in  Ex.  5. 

4.  The  Term  *'  Divisor."  In  certain  te.xts  it  is  stated  that  the  relation 
ai  a2  .  •  •  «n  =  ±Pn  in  (4)  implies  that  "every  root  of  an  equation  is  a  divisor 
of  the  absolute  term."  This  statement  is  either  trivial  or  else  is  not  always  true. 
It  is  trivial  if  it  means  merely  that  the  absolute  term  can  be  divided  by  any  root 
(that  root  being  a  complex  number),  yielding  a  quotient  wliich  is  a  complex  num- 
ber. For,  in  this  sense  division  is  always  possible  (except  when  the  divisor  is 
zero),  and  a  root  not  zero  is  a  divisor  of  any  numl^cr  whatever.  The  statement 
quoted  was  certainly  not  meant  in  this  trivial  sense,  with  no  special  force.  The 
only  other  sense,  familiar  to  the  reader,  in  which  a  constant  is  said  to  be  a  divisor 


5  5,  6]  THEOREMS  ON  ROOTS  OF  EQUATIONS  59 

of  another  constant  is  the  following:  An  integer  r  is  a  divisor  of  an  integer  p  if 
p/r  is  an  integer,  so  that  p  =  rq,  where  q  is  an  integer.  For  example,  4  is  a  divisor 
of  12,  but  not  of  6.  In  this  reasonable  sense  of  the  term  divisor  in  such  a  connec- 
tion, the  statement  quoted  becomes  intelligible  only  when  modified  to  read:  every 
integral  root  of  an  equation  with  an  integral  absolute  term  is  a  divisor  of  that 
term.  But  this  is  not  always  true.  The  integral  root  6  of  x'  —  -3°-  x  +  4  =  0  is 
not  a  divisor  of  4;  the  root  2  of  .c-  —  ^  a;  —  3  =  0  is  not  a  divisor  of  —3.  The 
correct  theorem  is  that  next  stated. 

5.  Integral  Roots.  For  an  equation  all  of  whose  coefficients  are  inte- 
gers, that  of  the  highest  power  of  the  variable  being  unity,  any  integral  root  is 
a  divisor  of  the  constant  term. 

In  certain  texts,  we  find  a  correct  statement  of  this  theorem,  but  an  erroneous 
proof.  When  ai  and  Pn  are  integers  and  aia2  .  .  .  an  =  ±  Pn,  it  is  falsely  con- 
cluded that  ai  is  a  divisor  of  />„.  But  12  •  3  •  j  =  9  and  12  is  not  a  divisor  of  9.  Also 
the  examples  at  the  end  of  §  4  show  the  falsity  of  this  argument  and,  indeed,  of 
any  argument  not  making  use  of  the  hypothesis  that  all  of  the  coefficients  are 
integers. 

A  correct  proof  is  very  easily  given.  Let  d  be  an  integral  root  of  equa- 
tion (1),  in  which  now  pi,  .  .  .  ,  /)„  are  all  integers.     Then 

(5)  d-  +  p,d"-'  +  Pid"-^  +     •    •    •     +  Pn-ld  +  Pn   =  0. 

Since  d  obviously  divides  all  of  the  terms  preceding  the  last  term,  it  must 
divide  p„. 

Hence  if  there  be  integral  roots  of  an  equation  of  the  specified  type, 
they  may  be  found  by  testing  in  turn  each  positive  and  negative  divisor 
d  of  the  constant  term  p„.  The  most  obvious  test  is  to  compute  (by  the 
abridgment  in  Ch.  I,  §  5)  the  value  of  f(d)  and  note  whether  or  not  this 
value  is  zero.  We  may  shorten  the  work  very  much  by  various  methods, 
and  most  by  a  combination  of  these  methods. 

Evidently  it  is  unnecessary  to  test  a  value  of  d  beyond  the  limits  of  the 
positive  and  negative  roots. 

6.  Newton's  Method  for  Integral  Roots.  Consider  an  equation  (1) 
with  integral  coefficients.  Let  d  be  an  integral  root.  It  is  a  divisor  of  pn 
and  we  may  set 

Pn  =  dq„-i. 

By  removing  the  factor  d  from  each  term  of  (5),  we  get 

d"-i  +  pid«-2  +  .  .  .  -f  p,^_^d  4-  p„_i  +  g„_i  =  0. 


60  THEORY  OF  EQUATIONS  [Ch.  vi 

The  left  member  is  divisible  by  d,  and  hence 

7J„_i  +  g„_i  =  dqn-2, 
where  g„_2  is  an  integer.     Then 

d"--  +  pid"-'  +     •    •     •     +  Pn--d  +  Pn-2  +  qn-2'=   0, 
Pn-2   +   qn-2   =   dqn-3, 

where  qn~z  is  an  integer,  etc.  Conversely,  if  such  a  relation  holds  at 
each  step  and  if,  finally,  1  +  go  is  zero,  then  df  is  a  root,  and  the  quotient 
of  f{x)  by  a;  —  d  is 

x"-^  —  qix"--  —  qoX"-^  _    .  .  .   _  q^_^x  -  g„_i. 

Indeed,  in  the  product  of  the  latter  by  x  —  d,  the  coefficient  of  .r"~'  for 
i  >  0  is  dqt-i  —  qt  and  this  equals  pt  by  our  relations. 

Corollary.  If  d  is  an  integral  root  of  an  equation  f(x)  =  x"  -\-  •  •  •  =  0 
with  integral  coefficients,  the  quotient  of  /(.r)  by  a:  —  d  is  a  polynomial 
with  integral  coefficients. 

This  process  is  a  modification  of  synthetic  division  (Ch.  X,  §4). 

Example,  fix)  =  .r*  —  9  x^  +  24  x-  —  23  .r  +  15  =  0.  Since  evidently  there 
is  no  negative  root,  and  since  10  is  an  upper  limit  to  the  positive  roots,  we  have 
only  to  test  the  divisors  1,  3,  5  of  15.  Now /(I)  =  8.  For  d  =  3,  the  work  is  as 
follows: 

1  -  9  24  -  23  15 

^  _6  -^  _5 

0  -  3  1-8  -  18 

Here  we  have  divided  15  by  3  and  placed  the  quotient  under  —  23.  Adding,  we 
get  —18,  whose  quotient  by  3  is  added  to  24,  etc.  Since  the  last  sum  is  zero,  3  is 
a  root.  The  (luotieiit  has  as  its  coefficients  the  negatives  of  the  numbers  in  tlie 
second  line  (see  the  first  line  below).     We  test  this  quotient  for  the  root  5: 

1  -6  6  -5 

-1  1  -  1 

0  -5  5 

Hence  5  is  a  root  and  the  quotient  is  x^  —  x  -\-  1.  The  latter  does  not  vanisli  for 
X  =  ±1.  Hence  3  and  5  are  the  only  integral  roots  and  each  is  a  sim])le  root. 
If  we  had  tested  a  divisor  —3  or  15,  not  a  root,  a  certain  (luotient  would  not  be 
integral  and  tlie  work  would  be  stopped  at  that  point. 

7.  Another  Method.  A  divisor  d  is  to  be  rejected  if  c^  —  m  is  not  a 
divisor  of /(m),  where  7ii  is  any  chosen  integer. 


§  81  THEOREMS  ON  ROOTS  OF  EQUATIONS  61 

For,  if  d  is  an  integral  root  of  f(z)  =  0, 

fix)  ^(x-d)  Q(x), 

where  Q(x)  is  a  polynomial  with  integral  coefficients  (§  6,  Cor.).     Then 

f(7n)  =  (m  —  d)q,  where  q  is  the  integer  Q{m). 

In  the  example  of  §  6,  /(I)  =  8  is  not  divisible  by  14,  so  that  15  is  not  an  integral 
root. 

Consider  the  new  example 

J{x)  =  x^  -  20 .1-2  +  164  X  -  400  =  0. 

There  is  no  negative  root  and  20  is  an  upper  limit  to  the  roots.  The  positive 
divisors  of  400  less  than  20  are  1,  2,  4,  16,  5,  8,  10.  The  last  three  are  excluded 
since /(I)  =  —255  is  not  divisible  by  4,  7,  or  9.  Also  16  is  excluded  since /(2)  = 
— 144  is  not  divisible  by  14.  Incidentally  we  have  excluded  the  divisors  1  and  2. 
The  remaining  divisor  4  is  seen  to  be  a  root  either  by  Newton's  method  or  by 
computing  /(4) . 

In  case  there  are  numerous  divisors  within  the  limits  to  the  roots,  it  is 
usually  better  not  to  begin  by  listing  all  of  the  divisors  to  be  tested.  For, 
if  a  divisor  is  found  to  be  a  root,  it  is  preferable  to  proceed  with  the  quo- 
tient, as  was  done  in  the  Example  in  §  6. 

EXERCISES 

Find  all  the  integral  roots  of 

1.  x^  -  10.c2  +  27.r-  18  =  0. 

2.  .r"  -  2  x^  -  21  x2  +  22  .c  +  40  =  0. 

3.  .t5  +  47  x*  +  423  x^  +  140  .c^  +  1213  x  -  420  =  0. 

4.  x^  -  34  x^  +  29  .r2  +  212  x  -  300  =  0. 

8.  Rational  Roots.  A7nj  rational  root  of  an  equation  with  integral 
coefficients,  that  of  the  highest  power  of  the  variable  being  unity,  is  necessarily 
an  integer. 

Let  a/b  be  a  root,  where  a  and  b  are  integers  with  no  common  divisor 
greater  than  unity.  Set  x  =  a/b  in  (1)  and  multiply  the  members  of  the 
resulting  relation  by  6"~^     We  get 

^  +  Pia"-^  +  P2a"-"b  +   •  •  •  +  Pn-\ab"-'~  +  p.b'^-^  =  0. 

All  of  the  terms  after  the  first  are  integers.  Hence  b  divides  a".  Unless 
b  =  ±1,  6  has  a  prime  factor  which  divides  a"  and  hence  also  a,  contrary 
to  hypothesis.     Thus  a/b  =  ±a  is  an  integral  root. 


62  THEORY  OF  EQUATIONS  [Ch.  vi 

The  rational  roots  of  any  equation  with  rational  coefficients  can  now 
be  readily  found.  If  I  is  the  least  common  denominator  of  the  fractional 
coefficients,  we  multiply  the  members  of  the  equation  by  I  and  obtain  an 
equation 

aoi/"  +  ai7/"-i  +  .  .  .  +  a„  =  0, 

where  ao,  .  .  .  ,  a„  are  integers.  Multiply  the  left  member  by  ao"~^  and  set 
Qoy  =  X.  We  obtain  an  equation  (1)  with  integral  coefficients,  that  of 
ic"  being  unity.  To  any  rational  root  ?/i  of  the  equation  in  ?/  corresponds 
a  rational  root  ao?/i  of  (1),  which  must  be  an  integer,  in  view  of  the  theorem 
just  proved.  Hence  we  need  only  find  all  of  the  integral  roots  of  the  new 
equation  (1)  and  divide  them  by  ao  to  get  all  of  the  rational  roots  y  of  the 
original  equation. 

Frequently  it  is  sufficient  (and  of  course  simpler)  to  set  ky  =  x,  where  k 
is  a  suitable  integer  less  than  ao. 

EXERCISES 

Find  all  of  the  rational  roots  of 

1.  y'  -  ¥  y'  +  ^F  f~  -  40  ?/  +  9  =  0. 

2.  6?/-  11 2/2  +  6 i/-  1  =  0. 

3.  108  2/3  -  270  2/2  _  42  2/  +  1  =  0.     [Use  k  =  6.] 

4.  32 2/3  -  6 2/  -  1  =  0.     [Use  the  least  k.] 

Form  the  equation  whose  roots  are  the  products  of  6  by  the  roots  of 

5.  a;2  -  2  X  -  i  =  0.  6-   a;^  -  i  x^  -  i  x  +  I  =  0. 


CHAPTER  VII 

Symmetric  Functions 

1.   S-polynomials ;  Elementary  Symmetric  Functions.    A  polynomial  in 

the  independent  variables  Xi,  Xo  ,  .  .  .  ,  Xn  is  called  symmetric  in  them  if 
it  is  unaltered  by  the  interchange  of  any  two  of  the  variables.     For  example, 

xi''  +  x~^  +  x^^  +  2,Xi  +  2>X2^Zxz 

is"  a  symmetric  function  of  xi,  Xo,  x^.  The  sum  of  the  first  three  terms  is 
denoted  by  Srcr  and  the  sum  of  the  last  three  by  3  S.Ci.  In  general,  if  t 
is  a  product  of  powers  ol  Xi,  .  .  .  ,  Xn,  whose  exponents  are  integers  S  0, 
Sf  denotes  the  sum  of  this  term  t  and  all  of  the  distinct  terms  obtained 
from  it  by  permutations  of  the  variables.  Since  such  a  S-polynomial 
2^  is  unaltered  by  every  permutation  of  the  variables,  it  is  unaltered  in 
particular  by  the  interchange  of  any  two  variables  and  hence  is  a  sym- 
metric function.     For  example,  if  there  are  three  variables  a,  /3,  y, 

Xa'^IS^y  =  a~^~y  +  cx^y'^  +  ^^y'~a, 

2a2/33^  =  a'^^'y  +  ^-c^y  +  a~y^(3  +  y'^a^^  +  jS^^a  +  y^^^a. 

Just  as  in  the  case  of  the  initial  example,  any  symmetric  polynomial  is 
evidently  a  linear  combination  of  S-polynomials  with  constant  coefficients. 
The  S-polynomials,  of  the  first  degree  in  each  variable, 

(1)  Ei  =  Sa:i,     E2  =  l^XiXo,     Ei  =  IiXiX-^Xs,  .  .  .  ,     En  =  XiXo  .  .  .  Xn-ix^ 

are  called  the  elementary  symmetric  functions  ol  Xi,  .  .  .  ,  Xn. 

Frequently  we  shall  employ  the  notation  cki,  .  .  .  ,  q;„  for  the  indepen- 
dent variables.  By  Ch.  VI,  §  1,  ai,  .  .  .  ,  q;„  are  the  roots  of  an  equation 
of  degree  n, 

(2)  f{x)  =  a;"  +  pio;"-!  +  pox""'^  •  •  •  -h  p„  =  0, 

in  which  —j)i,  po,  —pi,...,  (  —  !)"?)„  equal  the  elementary  sj^mmetric 
functions  of  the  roots.  It  is  customary  to  make  the  latter  statement 
also  for  an  equation  whose  roots  are  not  independent  variables. 

63 


64  THEORY  OF  EQUATIONS  (Ch.  VII 

But  in  the  latter  case  it  is  preferable  to  saj'  that  —pi,P2,  .  .  .  equal  the  elemen- 
tary symmetric  functions  formed  for  the  roots,  thus  indicating  that  we  have  in 
mind  the  values  of  certain  functions  of  arbitrary  variables  Xi,  .  .  .  ,  .t„  for  Xi  =  ai, 
.  .  .  ,  Xn  =  an-  It  may  happen  that  the  resulting  polynomials  in  ai,  .  .  .  ,  a„  are 
not  sj^mmetric  in  ai,  .  .  .  ,  «„.  For  example,  if  the  three  roots  are  a,  /3,  /3,  we  have 
—  pi  =  a  +  2  /3,  p2  =  2  a/S  +  /S',  —p3  =  CC0-,  which  are  the  values  of  Xi-\-  Xi  +  Xs, 
etc.,  but  are  not  themselves  symmetric  in  a,  /3,  /3,  being  altered  by  the  interchange 
of  a  and  ;8. 

However,  this  point  will  give  no  trouble  in  the  exercises  below,  since  the  roots 
are  given  distinct  notations  andjnay,  if  it  is  desu'ed,  be  regarded  as  independent 
variables. 

2.  Products  of  S-polynomials.  It  is  a  fundamental  theorem  that  any 
symmetric  polynomial  in  the  roots  is  expressible  rationally  and  integrally 
in  terms  of  pi,  pz,  .  .  .  ,  Pn  and  the  coefficients  of  the  symmetric  poly- 
nomial. To  prove  this,  it  suffices  to  show  that  any  2-polynomial  is  ex- 
pressible rationally  and  integrally  in  terms  of  the  elementary  symmetric 
functions.  Postponing  the  general  proof,  we  shall  now  treat  several  special 
cases  and  assign  others  as  exercises. 

Example  1.     If  a,  /3,  7  are  the  roots  of  x'  +  ?^-i'"  +  qx  +  r  =  0, 

p'-=  {a  +  l3+yy-  =  a-  +  fi-  +  y'-  +  2  (a/3  +  ay  +  fiy)  =   Sa^  +  2  q, 
Sa2  =  p2  _  2  g,       _  pg  =  2a  .  Sa^  =  Sa^^  +  3  a^y,      Zoc^^  =  3  r  -  pq, 
Xc^py  =  pr,      Sa2^2  ^  CEapy  -  2  a^yZa  =  q^  -  2  pr. 

The  student  should  carry  out  in  detail  the  steps  here  indicated. 

Example  2.  The  student  should  learn  how  to  express  a  product  like  Sa  •  Xa0 
in  Ex.  1  as  a  sum  of  2-functions  without  writing  out  their  expansions,  since  the 
latter  method  is  very  laborious  in  general.  To  obtain  the  types  of  2-functions 
in  the  product,  it  suffices  to  use  a  single  term  (called  leader)  of  one  factor,  saj''  a. 
Then  if  we  use  any  term  of  lla^  which  contains  a,  we  get  a  term  of  Z)a-/3;  while 
if  we  use  any  terra  not  containing  a  (hence  0y  in  this  example),  we  get  a  terra 
afiy.  It  remains  to  find  the  coefficients  of  these  2-functions  I2a-i3  and  a0y.  To 
get  0^13,  we  must  take  the  term  a  of  ^a  and  the  terra  a0  of  "Eaff,  so  that  1,a'0  has 
the  coefficient  unity.  To  get  a/37,  we  maj^  take  a  or  /3  or  y  from  2a  and  the  com- 
plementary factor  I3y  or  ay  or  a/S,  respectively,  from  2a/3.     Hence 

2a  .  2a^  =  'Ea'^13  +  3  a0y. 

3      3  6 

As  a  check,  we  have  marked  under  each  2  the  number*  of  its  terms.     Then  the 
total  number  of  terms  is  3  X  3  =  6  +  3. 

*  Found  by  the  theory  of  combinations  in  Algebra,  and  not  by  writing  out  in  full 
the  S-functions. 


S3]  SYMMETRIC  FUNCTIONS  65 

Example  3.    To  find  the  product  of  the  S-functions 

of  a,  (3,  7,  5,  we  use  the  leader  a.^  of  the  first.  To  obtain  the  four  types  of 
2-f unctions  in  the  product,  we  first  use  a  term  of  s  containing  both  a  and  /3; 
second,  a  term  of  s  containing  a-  but  not  /S;  third,  a  term  with  a  but  with  neither 
o^  nor  /3;  fourth,  a  term  free  of  a  and  /3.     The  respective  types  are  those  in 

2a,S  •  Za^3  =  1  ^a^0'  +  2  Za^/Jy  +  2  "^a^^y"-  +  3  ^cc^y'b. 

6        12  12  12  12  4 

The  coefficient  of  any  S-function  on  the  right  is  obtained  by  counting  the  num- 
ber of  ways  its  leader  can  be  expressed  as  a  product  of  terms  of  the  2-functions 
on  the  left. 

The  coefficient  of  ot^t  is  2  since  we  must  take  either  a/3  or  fiy  from  'Zafi  (for, 
we  must  take  a  or  y,  since  s  does  not  have  a  term  with  two  exponents  equal  to  2; 
while  if  we  take  ay,  the  complementary  factor  a^y  is  not  in  s).  To  obtain  afiy-b, 
we  must  take  a  term  from  s  with  7-  and  a  or  /3  or  5.  The  first  and  second  coefficients 
are  evidently  correct. 

EXERCISES 

If  a,  /3,  7,  5  are  the  roots  of  x^  +  px^  -\-  qx-  -\-  rx  -{-  s  =  0,  find 

1.  2a-/3-.    [Square  2a/3.] 

2.  'LoL^0.      [Use  Sa^  .  2a/3.] 

3.  Sa".       [Square  Za\] 

If  a,  /3,  7  are  the  roots  of  x^  +  p-C"  +  g.r  +  r  =  0,  find  the  cubic  equation  with 
the  roots 

2       2       2 

4.    a^,  ,32,  7'.  5.     a/3,  a7,  /37.  6.    -,      -,      -• 

a        p        7 

By  multiplying  2,ri  by  a  suitable  S-function,  express  in  terms  of  functions  (1) 
7.    Sxi2  (if  n  >  1).  8.   2.C12.C2  if  (n  >  2).  9.   1^x^X2  (if  n  =  2). 

10.    Sxi"  (if  71  >  2).  11.    2.C1--*  (if  n  -  2).  12.    llxiHiXz. 

13.  For  equation  (2)  with  n  >  4,  show  that 

2ai2a2a3a4  =  —  P1P4  +  5  Ps,      llctx^at^az  =  3  /:»i7)4  —  Jiip^  "  5  Ps- 

14.  For  equation  (2)  with  n  >  5,  show  that 

2ai2a2^a3a4  =  p2p4  —  4  piPs  +  0  Pe,      2ai2a2'''a3^  ==  P3^  —  2  ^2^4  +  2  piPs  —  2  J^e- 

3.  Fundamental  Theorem  on  Symmetric  Functions.  Any  polynomial 
symmetric  in  Xi,  .  .  .  ,  .r„  equals  a  polynomial  in  the  elementary  symmetric 
functions  Ei,  .  .  .  ,  En  of  the  x's. 

The  proof,  illustrated  in  Exs.  1  and  2  of  §4,  tells  us  just  what  elemen- 
tary symmetric  functions  should  be  multiplied  together  in  seeking  the 
expression  for  a  given  symmetric  polynomial  in  terms  of  the  £^'s  and 
hence  perfects  the  tentative  method  used  in  the  earlier  examples. 


66  THEORY  OF  EQUATIONS  [Ch.  vii 

It  suffices  to  prove  the  theorem  for  any  homogeneous  symmetric  poly- 
nomial *S,  i.e.,  one  expressible  as  a  sum  of  terms 

h  =  axi^^Xi'i  .  .  .  Xn''" 

of  constant  total  degree  k  =  h  +  ki  -\-  •  •  •  +  fc„  in  the  x's.  Evidently 
we  may  assume  that  no  two  terms  of  *S  have  the  same  set  of  exponents 
ki,  .  .  .  ,  kn  (since  such  terms  may  be  combined  into  a  single  one).  We 
shall  say  that  h  is  higher  than  the  term  hxi^x-,^'^  .  .  .  xj'n  if  ki  >  U,  or  if 
ki  =  li,  k'i  >  k,  or  if  fci  =  ^1,  A-2  =  k,  ks  >  k,  .  .  .  ,  so  that  the  first  one  of 
the  differences  ki  —  h,  ko  —  k,  ks  —  k,  .  .  .  which  is  not  zero  is  positive. 
If  the  highest  term  in  another  symmetric  polynomial  S'  is 

h'  =  a' x^^' X-]"''  .  .  .  a:/"', 

and  that  of  S  is  h,  then  the  highest  term  in  their  product  SS'  is 

hh'  =  aa'a;/i+^''  .  .  .  a:„'^"+'^""'. 

Indeed,  suppose  that  SS'  has  a  term,  higher  than  hh\ 

(3)  cxi''+'''   .  .  .  xj-+^-', 

which  is  either  a  product  of  terms 

t  =  bxi^'  .  .  .  Xn'",     t'  =  h'xi^^'  .  .  .  xj» 

of  S  and  S'  respectively,  or  is  a  sum  of  such  products.  Since  (3)  is  higher 
than  hh',  the  first  one  of  the  differences 

I  -\-  il     —   A"l   —   a"i  ,     .     .     .    ,    („  "T  tn  "n  '^"n 

which  is  not  zero  is  positive.  But,  either  all  of  the  differences  h  —  ki,  .  .  .  , 
In—  kn  are  zero  or  the  first  one  which  is  not  zero  is  negative^  since  h  is 
either  identical  with  t  or  is  higher  than  t.  Likewise  for  the  differences 
h'  —  ki,  .  .  .  ,  In  —  kn'.     We  therefore  have  a  contradiction. 

It  follows  at  once  that  the  highest  term  in  any  product  of  homogeneous 
symmetric  polynomials  is  the  product  of  their  highest  terms.  Now  the 
highest  terms  in  Ei,  Eo,  E3,  .  .  .  ,  En,  given  by  (1),  are 

Xi,         3-'iX2j       ^'l^2'r3,     .     .     .    ,       XiXi    .    .    .    Xn, 

respectively.     Hence  the  highest  term  in  Ei'^^E-f"-  .  .  .  £'„""  is 


§4]  SYMMETRIC   FUNCTIONS  67 

We  next  prove  that,  in  the  above  highest  term  h  of  >S, 

kl   =   A^2   =   ft'S    •     •     •    —   kn- 

For,  if  ki  <  k^,  the  symmetric  polynomial  S  would  contain  the  term 

axi^'^xj'^Xi^^  .  .  .  xj"", 
which  is  higher  than  h.     If  ko  <  h,  S  would  contain  the  term 

aXi'^^Xo'^^Xs''^  .  .  .  xj^", 
higher  than  h,  etc. 

By  the  above  result,  the  highest  term  in 

a  =  aEi'^^-^^E'i'^-''^  .  .  .  ^„_/-" '-^•"£;/« 

is  h.  Hence  Si  =  S  —  a  is  a,  homogeneous  symmetric  polynomial  of  the 
same  total  degree  k  as  S  and  having  a  highest  term  hi  not  as  high  as  h. 
As  before,  we  form  a  product  cri  of  the  E's  whose  highest  term  is  this  hi. 
Then  S2  =  Si  —  ai  is  a  homogeneous  symmetric  polynomial  of  total 
degree  k  and  with  a  highest  term  /12  not  as  high  as  hi.  We  must  finally 
reach  a  difference  St  —  (Ti  which  is  identically  zero.  Indeed,  there  is 
only  a  finite  number  of  products  of  powers  of  Xi,  .  .  .  ,  Xn  of  total  degree  k. 
Among  these  are  the  parts  h' ,  hi,  h^' ,  ...  of  /i,  hi,  ho,  .  .  .  with  the  coeffi- 
cients suppressed.  Since  each  hi  is  not  as  high  as  hi-i,  the  h',  hi,  h^ ,  .  .  .  are 
all  distinct.     Hence  there  is  only  a  finite  number  of  hi.     Since  St  —  at  ^  0, 

S   =  (J-\-Si   =   (T-\-iyi-{-So=     '    •    •     =  ff  +  CTl  +  (T2  +     •    •    •     +0"^ 

Hence  *S  is  a  polynomial  in  Ei,  Ei,  .  .  .  ,  En. 

4.  At  each  step  of  the  preceding  process,  we  subtracted  a  product  of 
the  £"s  multiplied  by  the  coefficient  of  the  highest  term  of  the  earlier 
function.  It  follows  that  any  symmetric  polynomial  equals  a  rational 
integral  Junction,  with  integral  coefficients,  of  the  elementary  symmetric  func- 
tions and  the  coefficients  of  the  given  polynomial. 

Corollary.  Any  symmetric  polynomial  with  integral  coefficients  can 
he  expressed  as  a  polynomial  in  the  elementary  symmetric  functions  with 
integral  coefficients. 

Instances  of  this  important  Corollary  are  furnished  by  the  results  in 
all  of  our  earUer  examples  and  in  those  which  follow. 


68  THEORY  OF  EQUATIONS  [Ch.  VII 

Example  1.    li  S  =  "^x^x^xz  and  w  >  4,  we  have 

«7  =  EiEz  =  5  +  3  2x12x2X3X4  +10  2x1X2X3X4X5, 

Si  =  S  —  a  =  —3  2X1^X2X3X4  —  10  2X1X2X3X4X5, 

(Ti  =  —  3  EiEi  =  —  3  (2x1^x2X3X4  +  5  2x1X2X3X4X5), 
'S2  =  Si  —  0-1  =  5  2x1X2X3X4X5  =  5  £'5, 
*S  =  cr  +  jSi  =  ff  +  tri  +  *S'2  =  £'2£'3  —  3  £'i£'4  +  5  £"5. 
Example  2.    If  *S  =  2x1^2X3  and  n  >  4, 

a  =  Ei^Ez  =  El  (2x1^x2X3  +  4  2x1X2X3X4) 

=  2x1^x2X3  +  2  2x1^X2^^X3  +  3  2xrX2X3X4 

+  4  (2Xi2x2.C3X4  +  5  2X1X2X3X4^-5), 

Si  =  S  —  (X  =  —2  2x12x2^X3  —  7  2xilr2X3X4  —  20  2x1X2X3X4X5. 

Take  o-i  =  —  2  £'2£'3  and  proceed  as  in  Ex.  1. 

Remark.     The  definition  of  a  2-polynoinial  in  §  1  may  be  extended  to  2-func- 
tions  in  general.     For  instance  if  there  are  three  variables  a,  0,  y, 

V- =  -  +  -  +  -.     V- =  -  +  -  +  -  +  -  +  -  +  -• 

^a        a        /3        7  ^a        a        a        0        P        y        y 

EXERCISES 

If  a,  /3,  y,  8  are  the  roots  of  x*  +  px^  +  qx-  +  ?'X  +  s  =  0, 

1.    y— -•  2.    V^  =  V«.Vl-4  =  ^-^'-4. 

Aa  S  A<x        A        ■Act  S 

3-2-.=--  4.    Vi  =  ^0-=-2s»). 

^^  or         ^^  ^^  or        ^^  a        S~ 

6.   Find  the  sums  in  E.xs.  1,  3,  4  from  the  smii,  sum  of  the  products  two  at  a 
time,  and  sum  of  the  squares  of  the  roots  of 

1  +  py  +  q>/  +  rif  +  si/  =  0, 

obtained  by  replacing  x  by  1/y  in  the  former  quartic  equation. 

0  +  y  +  d       ^-p  -  «  .  x^l 
=  > =  -4.-P  2,-' 

_V^  o     V  7   _  3  r  -  p(7 

12 


7. 

12                  4 

8. 

'O  «'  +  /S" 

10. 


SYMMETRIC  FUNCTIONS  69 

12.  Prove  that  the  degree  in  any  single  x  of  a  homogeneous  symmetric  poly- 
nomial *S'  is  the  total  degree  of  the  equal  polynomial  in  the  E's.  Hints:  First  show 
that  no  term  of  *S'  has  an  exponent  >  A;i,  so  that  the  degree  of  S  in  any  single  x 
is  ki.  Next,  o-  is  of  total  degree  A;i  in  the  £"s.  Set  h  =  a'xi'^  ....  Then  <7i 
is  of  total  degree  A-i'(  =  A:i)  in  the  £"s  and  not  every  exponent  in  o-i  equals  the  corre- 
sponding exponent  in  a.     Thus  a  is  not  cancelled  by  o-i,  ai,  .  .  .  . 

13.  Given  a  pol.ynomial  in  the  £"s  of  total  degree  d,  show  that  the  equal  func- 
tion of  the  .t's  is  of  degree  =  d  in  any  single  root. 

5.   Sums  of  Like  Powers  of  the  Roots.     If  ai,  .  .  .  ,  «„  are  the  roots  of 

(2),  we  write  Si  =  2ai,  So  =  Soii^,  and,  in  general, 

Sk  =  ^ai    =  a-^  +  oi'i'  +   •  *  *   +  oCt!'- 
The  factored  form  of  (2)  is 

(4)  j{x)  =  {x  -  a^{x  —  a^^    .   .   .   {x  -  an). 

In  this  identity  in  x,  we  may  replace  x  hy  x  -{■  h.     Thus 

J{x  +  h)  =  (x  +  /l  —  ai){x  -\-h  —  a'^^   .   .   .   (x  -\-  h  —  an). 

In  the  expansion  of  f{x  +  h)  as  a  polynomial  in  h,  the  coefficient  of  the 
first  power  of  h  isf'{x),  by  the  definition  of  the  first  derivative  oi  f{x)  in 
Ch.  I,  §  4.     In  the  right  member,  the  coefficient  of  h  is 

(x  -  a-i)(x  —  as)    .   .   .    {x  —  a„)+    •  •   •    +(x  —  ai)(x  —  a-z)  .   .   .  (x  —  q:„_i). 

Here  the  first  product  equals /(a;)  -^  {x  —  ai),  by  (4),  etc.     Hence 

(5)  /'(;,)  ^JM_  +  JM_+         .    /(^) 


X  —  ai       X  —  a2  X  —  an 

If  a  is  any  root  of  (2) ,  /(a)  =  0  and 

f(x)     _  f(x)  -  /(«)  _  a:"  -  g"  ^.n-i  _  ^n-i  x  —  a 

~.  „  I     /  1  i~    *    *    ■       \     yn—l 

X  —  a  X  —  a  X  —  a  X  —  a  x  —  a 

=  x"--^  +  cxx"-~  +  a'^x"-^  +  •  •  •   +  pi(a;"-2  -f  ocx'''^  +  •  •  •  ) 

(6)      ^^  =  x^-'  +  (a  +  pi)x"-2  +  (a2  +  p,a  +  po)^:"-^  +   •  •  • 

+  (a''  +  pia*^-i  +  poQ:*-2  +    .  •   .    +  p,^_,a  -\-  Pk)x''-''-^  +    '   •  '  . 
Taking  a  to  be  ai,  .  .  .  ,  «„  in  turn  and  adding  the  results,  we  have  by  (5) 
f'{x)  =  wx"-i  +  (si  -f  npi)x"-^  +  (s2  +  piSi  -\-  np2)x''~^  ■  ■  • 

-\-  {Sk  -\-piSk-i-\-  P2Sk-2  +  '  ■  •   +  PA;-iSi  +  np;t)a:"-*-i+  •  •  •  . 


70  THEORY  OF  EQUATIONS  ICh.  vil 

By  Ch.  L,  §  4, 

f'{x)=nx"-'-\-{n-l)piX"-^-\-{n-2)p2X"-^-{-  •  •  •  -\-{n-k)pkX--''-'-{-  •  •  •» 
Since  the  coefficients  of  like  powers  of  x  are  equal,  we  get 

(7)  si  +  pi  =  0,     So  +  piSi  +  2  p2  =  0,  .  .  .  , 

Sk  +  ViSk-i  +  P2Sfc-2  +  •  •  •  +  P;fc-iSi  +  kpk  =  0     (A;  =  1,  2,  .  .  .  ,  w  -  1). 
We  may  therefore  find  in  turn  Si,  So,  .  .  .  ,  s„_i : 

(8)  Si  =  -2h,     S2  =  pi^  -2  p-i,     S3  =  -pi^  +  3  pipo  -  3  p3,  .  .  .  . 

To  find  s„,  replace  a;  in  (2)  by  ai,  .  .  .  ,  a„  in  turn  and  add  the  resulting 
equations.     We  get 

(9)  Sn  +  p-.Sn-l  +  P2Sn-2  +     '     '     "     +  Pn-l^l  +  lipn   =   0. 

We  may  combine  (7)  and  (9)  into  a  single  formula: 

(10)  Sk  +  piSk-i  +  PiSk-o  +  •  •  •  +  Pk-iSi  +  kpk  =  Q    (k  =1,2,  .  .  .  ,n). 

To  derive  a  formula  which  shall  enable  us  to  compute  the  Sk  for  k  >  n, 
we  multiply  (2)  l)y  a;^"",  take  x  =  ai,  .  .  .  ,  x  =  an  m  turn,  and  add  the 
resulting  equations.     We  get 

(11)  Sk  +  PiSk-i  +  PiSk-i  +  •  •  •  +  PnSk-n  =  0  (k  >  n). 

Relations  (10)  and  (11)  are  called  Newton's  form  uIob.  They  enable  us 
to  express  any  Sk  as  a  poljoiomial  in  pi,  .  .  .  ,  Pn- 

EXERCISES 

1.  For  a  cubic  equation,  S4  =  2)1''  —  4  pi-p2  +  4  pips  +  2  7)2^. 

2.  For  an  equation  of  degree  n  S  4,  S4  =  pi^  —  4  j^rpo  +  4  ]h]h  +  2  7J2-  —  4  ^4. 

3.  If  we  define  Pn+i,  Pn+2,  ...  to  be  zero,  relations  (10)  hold  for  every  k. 
Hence  if  .pi,  p2,  .  ■  .  are  arbitrary  numbers  unlimited  in  number,  and  if  o-i,  0-2,  ..  . 
are  computed  by  use  of 

<^fc  +  Pi<rk-i  4-  •  •  •  +  Pk-i(n  +  kpk     {k  =  1,2,...), 

(Tk  becomes  Sk  when  we  take  pn+i  =  0,  p„+2  =  0,  .  .  .  .     See  Exs.  1,  2. 

4.  For  x"  —  1  =  0,  Sk  =  n  or  0  according  as  A;  is  divisible  or  not  by  n. 

6.   2-f unctions  Expressed  in  Terms  of  the  Functions  s,,.     We  have 

SaSb  =  -0:1"  •  -ai*"  =  2:0:1''+''  +  m  :^afa-fi, 

(12)  Sai''a2^   =  -  (SaSb  -  Sa+b), 

where  m  =  1  if  a  5^  5,  m  =  2  \i  a  =  b. 


§  7]  SYMMETRIC  FUNCTIONS  71 

Any  S-f unction  with  a  term  involving  just  three  roots  may  be  denoted 
by  Sai^ao^'as",  a  g  6  ^  c  >  0.     If  6  >  c, 

for  7n  as  above.     Since  a  -\-b  >  c,  a  -{-  c  >  b, 

mllafa-^a-f=   Sa  {SbSc  —  Sh+c)  —  {Sa+bSc  —  Sa+6+c)  —  (Sa^-cS^  —  Sa+h+c), 

(13)  '^afa'^a:^   =  —  (SaSbSc   —  SaSb+c   —   SbSa+c   —  ScSa+b  +  2  Sa+b+c)     (6  >  c). 

But  if  6  =  c,  we  have 
where  r  =  1  if  a  >  6,  r  =  3  if  a  =  6.     Hence 

(14)  'Eai'^a'J'as^   =   K^aSb'  -  S„S2  6  -  2  SbSa+b  +  2  Sa+2  6)  (tt   >   6), 

(15)  Zai^aa^as"  =  h(sj  -  3  s^Sa  a  +  2  S3  a). 

The  fact  that  any  '  "Z-polynomial  can  be  expressed  as  a  polynomial  in  the 
functions  Sk  is  readily  proved  by  induction.     We  have 

Sa^ai^ai"  .  .   .  ar^  =  f^afao^a^'  .   .   .  a^+i"  +  ti'Eai^+^ao''  .  .   .  ar" 

+   •  •  •   +  tri:,a^a2''  .   .   .  0:/+^, 

where  t  is  a  positive  integer,  and  ti,  .  .  .  ,  tr  are  integers  s  0  (for  example, 
<r  =  0  if  gf  =  6,  since  the  terms  which  it  multiplies  are  included  in  the  sum 
multiplied  by  ti).  Hence  if  every  :Sa:i^'  .  .  .  aj^r  is  expressible  as  a  poly- 
nomial in  the  functions  Sk,  the  same  is  true  of  every  '^ai^a-^  .  .  .  0:^+1^.  But 
the  theorem  is  true  for  r  =  1  (by  the  definition  of  s^.) .  Hence  it  is  true  by 
induction  for  every  r. 

EXERCISES 

1.  Take  a  =  6  in  (13)  and  then  replace  c  by  a.  Hence  (14)  holds  also  when 
a  <  b.     Derive  this  result  just  as  we  did  (14). 

2.  Express  2  (xx°-aiaz''ai''  in  terms  of  the  Sk,  treating  all  cases.  Why  are  these 
formulje  unnecessary  if  the  equation  is  of  degree  four? 

3.  For  a  quartic  equation  express  the  functions 

2/ai''a2  ,      ^ai  a2,       ~iai~ataz,^    Zj(xi"ai-az 

in  terms  of  the  s^  and  ultimately  in  terms  of  the  pi,  .  .  .  ,  p4. 

7.  Since  any  Sk  equals  a  polynomial  in  pi,  .  .  .  ,  p„(§  5),  the  theorem 
of  §  6  shows  that  any  2-polynomial  (and  hence  any  rational  integral 
symmetric  function)  of  the  roots  of  an  equation  equals  a  polynomial  in 


72  THEORY  OF  EQUATIONS  tCn.  vil 

the  coefficients  pi,  .  .  .  ,  pn  of  the  equation.  Since  we  may  form  an  equa- 
tion with  arbitrarily  assigned  roots,  we  have  a  new  proof  of  the  funda- 
mental theorem  on  symmetric  functions  (§3). 

The  method  of  §§  5,  6  to  express  a  S-polynomial  in  terms  of  the  coeffi- 
cients is  advantageous  when  a  term  of  2^  involves  only  a  few  distinct 
roots,  but  with  high  exponents,  while  the  method  of  §§  2,  3  is  preferable 
when  a  term  of  S  involves  a  large  number  of  roots  with  low  exponents. 

8.  Waring's  Formula  *  for  s^  in  Terms  of  the  Coefficients.  We  shall 
first  derive  this  formula  by  a  very  brief  argument  employing  infinite 
series  in  a  complex  variable,  and  later  give  a  longer  but  more  elementary 
proof. 

In  (2)  and  (4)  replace  x  by  \/y  and  multiply  each  by  ?/".     Thus 

(16)  1  +  piU  +  p^if  +   •  •  •    +  pnV"  =(1  -  «i2/)(l  -  aiij)  .  .  .  (l-anV). 

Take  the  natural  logarithm  of  each  member,  noting  that  the  logarithm 
of  a  product  equals  the  sum  of  the  logarithms  of  the  factors,  and  that 

1  00       ^ 

log  (1  -  2)  =  -2  -  I  2^  _   1  23  _     .    .    .     -  -zr  -   .    .    .    =   _  ^  _  ^r^ 

if  the  absolute  value  of  2  is  <  1.     Hence 

-X(-i)^|-.(pi2/+  •  •  •  -^Pn>ry=  -%lw-]-  ■ . .  +a/)r 

r=l  r=l 

00        ■, 

if  y  is  sufficiently  small  in  absolute  value  to  ensure  the  convergence  of 
each  of  the  series  used.      The  coefficient  of  y^"  in  (piy  +  •  •  •  +  Pny'^Y 
may  be  found  by  the  multinomial  theorem.     Hence,  after  dividing  r  =  ri  + 
•  •  •  +  r„  into  the  multinomial  coefficient,  we  get 

,,.,,  ^  (-!)'•■+ •••+'-..A;-(ri+  •  •  •   +  r„  -  1)!     ^     ^ 

(10     s,^^ ;,!  ,,!...  ,„! T'^'P^'  •  •  •  P'^^ 

*  Edward  Waring,  Misc.  Analyl.,  1762;  Meditationes  Algehraicce,  1770,  p.  225,  3d  ed., 
1782,  pp.  1-4.  No  hint  is  given  as  to  how  Waring  found  (17);  his  proof  was  in  effect 
by  mathematical  induction,  being  a  verification  that  Sk,  sa— i,  .  .  .  ,  Si  satisfy  Newton's 
formulip. 

But  (17)  had  been  given  earUer  by  Albert  Girard,  Invention  nouvelle  en  I'alghbre 
Amsterdam,  1G29. 


§91  SYMMETRIC  FUNCTIONS  73 

where  the  sum  extends  over  all  sets  of  integers  ri ,  .  .  .  ,  r„,  each  S  0,  for 
which 

(18)  ri  +  2  r2  +  3  rs  +  •  •  •  -{- nVn  =  k. 
Here  r!  denotes  1  •  2  •  3  .  .  .  r  if  r  S  1,  and  unity  if  r  =  0. 

9.   Elementary  Proof  of  Waring's  Formula.     Divide  each  member  of 
(16)  into  the  negative  of  its  derivative;  we  get 

(19)  -pi-2  pojj  -   ■  ■  '   -  npnV''-^  _       ai       _,_...   _^ 


1  +  Pi2/  +  •  •  •  +  PnU"  1  -  aiy  1  -  any 

In  the  identity 

(20)  T^^i  +  Q  +  Q"+  •  •  •  +Q'-'+    ^^ 


l-Q"      i  ^  :  ^    .  ■  ^        '  l-Q 

set  Q  =  agy  and  multiply  the  resulting  terms  by  ag.  Hence  the  second 
member  of  (19)  equals 

(21)  .  +  ,.,+  ...+  ..,/-.  +  .^^,^f/^'.^^^^., 

the  polynomial  (f)(y)  being  introduced  in  bringing  the  fractional  terms 

ai^+V(l  -  «i2/), 
etc.,  to  the  common  denominator  (16). 

In  (20),  we  now  set  Q  =  —piy  —  •  •  •   —  Pn?/".     Thus 

!  +  ,..+ '••+P.r  -  2;  (-  mp^y  +■■■+  v.ry  +  YT^-' 

r=U 

where  \p{y)  is  a  polynomial.  Expanding  this  rth  power  by  the  multino- 
mial theorem,  we  see  that  the  left  member  of  (19)  equals 

^y(-l)r,+  ...+r„  +  l(^l+  '".^"''•pi'-i  •    •   •  p/nyn+^r,+  ..-+nr„_^E 

^^      '  ri!  •  •  •  r„! 

(d  =  Pi+2p22/4 ), 

the  sum  extending  over  all  integral  values  ^  0  of  ri,  r2,  .  .  .  ,  rn  such 
that  ri  +  •  •  •  +  ^re  <  fc,  while  ^^  is  a  fraction  whose  denominator  is 
1  +  P\y  +  •  •  •  and  whose  numerator  is  the  product  of  y^  by  a  polynomial 
in  y.  In  the  expansion  of  the  part  preceding  E,  the  terms  with  the  factor  y^ 
may  be  combined  with  E  after  they  are  reduced  to  the  same  denominator 


74  THEORY  OF  EQUATIONS  [Ch.  vil 

as  E.     The  resulting  expression*  is  now  of  the  same  general  form  as  (21), 
so  that  the  coefficient  of  /"'  must  equal  Sk.     This  coefficient  is  the  sum  of 

^^_iy.+  •  •  •  4-.„  +  i(^^  +  •  •  •  +'"^-p/-+ V-  .  .  .  P/" 

(ri  +  2r2+  •  •  •   +nrn  =  k-  1), 

2;^(-l)n+  ■  •  •  +r„+i(^^  +   •  •  •   +  ""^-p/.p/.+  i  .   .   .  pj. 

(ri  +  2  ro  +  •  •  •   +  nr„  =  A;  -  2), 

3^(_l)n+...+.„+i^^'^+   •  •  •   +J^'^-p,r.pj^-p/,+  i  .  .  .  p„r„ 

(n  +  2  ro  +  .  .  .  +  nr„  =  A;  -  3), 


In  the  first  sum  employ  the  summation  index  Vi  +  1  instead  of  ri;   in 
the  second  sum,  r-z  +  1  instead  of  ry]  etc.     We  get 


(ri  +  •  •  •   +  r„  -  1) 


^  ri!r2!  (rs  —  1)!  .  .  .  r„! 


where  now  (18)  holds  for  each  sum.     Adding  these  sums,  we  evidently 
get  the  second  member  of  (17). 

Example  1.    Let  ?t  =  3,  A;  =  4.    Then  ri  +  2  r2  +  3  ra  =  4  and 
(ri,  r2,  r3)  =  (4,  0,  0),     (2,1,0),     (1,0,1),     (0,2,0), 
/3'  2'  1'  1'      \ 

=  ]h*  -  4  pi-p2  +  4  PiP3  +  2  P2'. 

*  The  difference  between  it  and  (21)  is  an  expression  of  the  form  (21).  Suppose  there- 
fore that  an  expression  (21)  is  identically  zero.  Taking  y  =  0,  we  get  6i  =  0.  The 
quotient  by  y  is  identically  zero.     Then  62  =  0,  etc. 


§  10]  SYMMETRIC  FUNCTIONS  75 

Example  2.  Let  n  =  2  and  write  p  for  —pi,  q  for  p2,  r  for  Vi.  Then  n  =  k  —  2r. 
If  /c  is  the  largest  integer  =  k/2,  the  suni  of  the  A;th  powers  of  the  roots  of 
X-  —  px  -\-  q  =  0  \s 

^{-lYk-jk-r-iy.    ,_^_ 
^^^A ik-2r)lrl ^       ^ 

r=0  ^  -^ 

10.  t  Certain  Equations  Solvable  by  Radicals.  Regarding  p  as  a  vari- 
able and  g  as  a  constant,  denote  the  polynomial  in  the  preceding  Ex.  2 
by  F(p).  The  equation  F(p)  =  c,  where  c  is  an  arbitrary  constant,  can  he 
solved  by  radicals.  Indeed,  if  x  is  a  particular  root  oi  x^  —  px  -{-  q  =  0,  the 
second  root  is  q/x,  and 

Sk  =  a;^  +  p    • 
\x/ 

This  expression  in  x  is  therefore  the  result  of  replacing  p  by  x  -\-  q/x  in 
F(p),  as  shown  by  the  quadratic  equation.     Hence  F{p)  =  c  then  becomes 

x''  -\-  (-]  =  c,     x-'^  —  cx^  +  5^  =  0. 

Solving  this  as  a  quadratic  equation  for  x^,  we  get 


^•---y/'^--^ 


^        2=^V4-5 


Since  the  product  of  these  two  expressions  is  q^,  definite  values 


i+v/f-'A  '=\7i-V3-«' 


can  be  chosen  so  that  pa  =  q.  Hence  if  e  be  a  primitive  kth  root  of  unity, 
the  2k  values  of  x  can  be  separated  into  pairs  pe™,  (re*=~^  (w  =  0,  1,  .  .  .  , 
k  —  I),  such  that  the  product  of  the  two  in  a  pair  is  pa  =  q.  Now  x  +  q/x 
is  a  value  of  p.     Hence  the  k  roots  p  of  F{p)  =  c  are 

p^m  _|_  ^^k-ni  (fn  =  0,  I,    .    .    .   ,   k  -  1). 

Thus  F{p)  =  c  can  be  solved  by  making  the  substitution 

I   Q 
p  =  x  -\-  -' 

X 

For  fc  =  3,  the  equation  is,  p^  —  Z  qp  =  c  and  the  present  method  be- 
comes that  in  Ch.  Ill  for  solving  a  reduced  cubic  equation. 


76  THEORY  OF  EQUATIONS  [Ch.  vil 

EXERCISES 

l.f  Solve  DeMoivre's  quintic  /;^  —  5  qp^  +  5  q-p  =  c  for  p. 

2.t  Solve  7;^  —  4  qp'^  +  2q^  =  c  for  /;  by  this  method. 

3.t  Write  down  a  solvable  equation  of  degree  7.     Solve  it. 

4.t  Solve  7/^  +  10  //  +  20  7/  +  31  =  0. 

11.   Polynomials  Symmetric  in  all  but  One  of  the  Roots.     If  P  is  a 

polynomial  in  the  roots  of  an  equation  f{x)  =  0  of  degree  n  and  if  P  is  sym- 
metric in  n  —  I  of  the  roots,  then  P  equals  a  polynomial  in  the  remaining  root 
and  the  coefficients  of  P  and  f{x) . 

For  example,    P  =  3  ai  +  ao-  +  as'-  +   •  •  •  +  a,c    is  such  a  polynomial  and 

P  —  2a;i-  +  3  ai  —  ar  =  pr  —  2  p-z  +  3  ai  —  ar. 

If  a  is  the  remaining  root,  P  is  symmetric  in  all  of  the  r(3ots  of  the  equa- 
tion (6)  of  degree  n  —  1,  whose  coefficients  are  polynomials  in  a,  pi,  .  .  .  , 
Pn.  Hence  (§  3)  P  equals  a  polynomial  in  a,  pi,  .  .  .  ,  pn  and  the  coeffi- 
cients of  P. 


Example  1.     If  a,  /3,  7  are  the  roots  of /(x)  s  x^  +  px-  +  qx  -\-  r  =  0,  find 

Xor  +  /3-  _  «-  +  0'   .   «-  +  7-   ,   jS'  +  7' 
a  +  13  a  +  /3  "  +  7  /3  +  7 

Since  ffi-  +  y-  =  p-  —  2  q  —  a-,      (3  +  y  =  —p  —  a, 

^a-  +  /3'-       ^;;--25-a'       ^/  ,2r/\  o.o^l 


But  a  -{-  p,  0  -\-  p,  y  -\-  p  are  the  roots  jji,  y^,  y%  of  the  cubic  equation  obtained  from 
j{x)  =  0  by  setting  x  +  p  =  y,  i.e.,  x  =  y  —  p.     The  resulting  equation  is 

if  _  2  py2  _^  (^j2  _^  ^^),y  -I-  ,.  -  pq  =  0. 

Since  we  desire  the  sum  of  the  reciprocals  of  ?/i,  7/2,  7/3,  we  set  7/  =  1/z  and  find  the 
sum  of  the  roots  Zi,  Zi,  za  of 


1  -  2  7^2  +  (/j2  +  ^)22  _,_  (,.  _  p^),3  =  0. 
Hence 

^a-jf^  _  2q-  -2  phj  +  4  pr 
+  /3  pq-r 


"  +  P       -^^i       ^^'       pq-r'        ^  <x 


§  111  SYMMETRIC  FUNCTIONS  77 

Example  2.     If  xi,  .  .  .  ,  Xn  arc  the  roots  of  f{x)  =  0,  find 


i'a;,  4 


Xi  +  C 

First,  Xi  +  c  =  ?/i,  .  .  .  ,  Xn  -\-  c  =  ijn  are  the  roots  of 

A-c  +  y)  =K-c)+yfi-c)+yK  )+...=  0. 
Next,  1/yi  =  Zi,  .  .  .  ,  1/yn  =  z,i  are  the  roots  of 

z"fi-  c)  +  z"-ri-  c)  +  z-\  )  +  .  .  .   =0, 
obtained  by  setting  y  =  l/z  in  the  preceding  etiuation.     Hence 

-n-c) 


-^  a;,  +  c      ^"^ 


Xy  +  c      ^  fi-c) 

EXERCISES 

[In  Exs.  1-14,  a,  0,  y  are  the  roots  of/(.c)  =  .r'  +  px"^  -{-  qx  +  r  =  0.] 

1.   Find  7  — ; — -1)7  means  of  the  last  formula. 

Using  /3y  +  q;(/3  -\-  y)  =  q,  find 

/37  +  «^  ^     ^  3  /3y  -  2  a2 


2.  y  ^.  , 

4.  Why  would  the  use  of  ^y  =  —r/a  complicate  Exs.  2,  3?    Verify 

-r      /■(«)  -  r 
0y  =  =  ' =  a-  +  pa  +  q. 

(X  <x 

5.  Why  would  you  use  Py  =  —r/a  in  finding  7 , ■? 

■*-/       ^y 

6.  Show  that  the  last  sum  equals  S(7//3). 

7.  Find^(/3  +  7)^         8.   Find  ^  («  +  ^  -  7)^        9.   Find  ^  f^-^^^- 

10.  Find  a  necessary  and  sufficient  condition  on  the  coefficients  that  the  roots, 

112  —3  r 

in  some  order,  shall  be  in  harmonic  progression.     If  — | —  =  -,  then /?  =  0, 

a      7      /3  q 

and  conversely.    But 

11.  Find  the  cuinc  ('(juation  with  the  roots  /S7 ,    ay ,   a/3 .     Hint: 

a  (3  y 

since  these  are  (  — r  —  l)/a,  etc.,  make  the  substitution  (— r  —  l)/x  =  y. 

Find  the  substitution  which  replaces  the  given  cubic  equation  by  one  with  the 
roots 


78  THEORY  OF  EQUATIONS  [Ch.  vii 

12.  a/3  +  ay,      a0  +  0y,      ay  +  0y. 

13.  — , ,  etc.  14.   — ■ -— ,etc. 

P  -\-  y  —  a  P  -\-  y  —  Z  a 

If  a,  13,  y,  d  are  the  roots  of  x*  +  px^  +  qx"^  -\-  rx  -\-  s  =  0,  find 

■    ^    ^+7  +  5   '  ■    ^/3  +  7  +  5-3' 

17.  If  2/1,  ?/2,  2/3  are  the  roots  of  y^  +  py  +  q  =  0,  the  equation  with  the  roots 
2i  =  (2/2  -  2/3)',     22  =  (2/1  -  2/3)^     23  =  (2/1  -  2/2)-  is 

2^  +  6  p2-  +  9  p-2  +  4  2^3  +  27  g-  =  0. 

Hints:  since  Zi  =  S/yr  —2  2/22/3  —  2/1^  =  — 2  p  +  2  g/2/1  —  2/1^,  etc.,  we  set  z  = 
— 2  p  +  2  5/2/  —  2/^.  By  the  given  equation,  ?/  +  ?>  +  g/2/  =  0.  Thus  the 
desired  substitution  is  2  =  —  p  +  3  q/y,     y  =  3  q/{z  +  />). 

18.  Hence  find  the  discriminant  of  the  reduced  cubic  equation. 

124  Cauchy's  Method  for  Symmetric  Functions.  If  Xi,  .  .  .  ,  Xn  are 
the  roots  of  (2),  any  polynomial  P  in  Xi,  .  .  .  ,  Xn  can  be  expressed  as  a 
polynomial  in  X2,  .  .  .  ,  Xn,  Pi,  .  .  .  ,  Pn,  in  every  term  of  which  the  expo- 
nent of  Xo  is  less  than  2,  the  exponent  of  X3  less  than  3,  .  .  .  ,  the  exponent 
of  Xn  less  than  n.  To  this  end,  we  first  eliminate  Xi  by  using  2.ri  =  —pu 
Then  we  eliminate  Xo'^'ik  =  2)  by  using  the  quadratic  equation  satisfied  by 
X2  and  having  as  coefficients  polynomials  in  Xs,  .  .  .  ,  x„.  This  quad- 
ratic may  be  obtained  by  dividing  f{x)  by  {x  —  X3)  .  .  .  (x  —  Xn),  or  by 
noting  that 

Xl  '^  X>   =    —pi   —  X3   —     •     •    •     —   Xn, 
3:10:2   =  Po-   {Xi  +  X2)  (.T3  +    •    •    •     +  Xn)   -  XzXi  -     ...     -  Xn-\Xn. 

Next,  we  eliminate  xz''{k  ^  3)  by  using  the  cubic  equation  obtained  by 
dividing /(x)  by  {x  —  x^  .  .  .  {x  —  a:„).  Finally,  we  eliminate  a:„*(A;  S  n) 
by  using  j{xn)  =  0. 

Ex-\MPLE.     To  compute  by  this  method  the  discriminant 

A  =  (xi  -  .r2)''(.ri  -  .r3)-(.r2  -  x^)- 
of  f{x)  3  x^  +  px  -\-  q  =  0,  we  note  that  Xi  and  X2  are  the  roots  of 

-^^  =  Qix)  =  x2  +  XX3  +  X32  +  p  =  0. 
x  —  X3 

Since  Sxi  =  0, 

(xi  -  X2)-  =  {-2x2-  XzY'  =  4  Q(X2)  -  3 .1-3-  -  4  p  =  - 3  X3-  -  4  />, 

(xi  -  Xi)(.C3  —  X2)  =  Q{x3)  =  3  X32  +  p, 

A  =  (-3  X32  -  4  p)(3  X32  +  pY  =  -27(.C3^  +  pxj)-  -  4  p', 

A=  -21q^-^p\ 


§131  SYMMETRIC  FUNCTIONS  79 

We  can  now  easily  prove  the  fundamental  theorem  of  §  3 :  if  P  is  sym- 
metric in  xi,  .  .  .  ,  Xn,  it  equals  a  polynomial  in  pi,  .  .  .  ,  pn-  For, 
P  =  A  -{-  Bxo,  where  neither  A  nor  B  involves  Xi  or  X2.  Since  P  is  unal- 
tered when  Xi  and  Xo  are  interchanged, 

A-\-Bx2  =  A+  Bxi. 

If  Xi  7^  Xi,  then  5  =  0;  and,  by  continuity,  B  —  0  even  when  Xi  =  Xz.    Hence 

P  =  C-^Dx,  +  Ex^, 

where  C,  D,  E  do  not  involve  a^i,  X2  or  x-i.  Since  P  is  unaltered  when  X3 
and  Xi  are  interchanged,  or  when  x^  and  X2  are  interchanged,  the  equation 

0  =  C-P-\-Dy  +  Ef- 

has  the  three  roots  Xi,  X2,  X3.  Hence  if  Xi,  X2,  Xs  are  distinct,  D  =  E  =  0, 
P  =  C,  and  by  continuity  these  relations  hold  also  if  two  or  all  three  of 
these  a;'s  are  equal,  so  that  P  is  free  of  Xi,  X2,  X3.  Similarly,  P  can  be  re- 
duced to  a  form  which  is  free  of  each  Xi. 

13.  t  Tschirnhausen  Transformation.  We  can  eliminate  x  between  (2) 
and 

(22)  X  =  uo  +  Uix  +  ihx"  +  ■  •  •  -i-Un-iX"-^ 

and  obtain  an  equation  in  X  of  degree  n.  First,  from  the  expressions  for 
X^,  X^,  .  .  .  ,  we  eliminate  x",  x"-+^,  ...  by  use  of  (2)  and  get 


(23) 


X^  =  W20  +  U21X  +  U22X-  -\-  •  •  •  -^  ih  n-i^;"  S 


X"   =  UnO  +  UnlX  -\-t('n2X^  +     •    '    '     +  ^^.  n-lX"'^, 


where  the  w,-,-  are  polynomials  in  Uo,  .  .  .  ,  Un-i  and  the  coefficients  of  (2). 

In  any  one  of  these  equations  (23)  we  set  x  =  a;i,  .  .  .  ,  x  =  Xn  in  turn 

and  add  the  resulting  relations.     If  Xi,  .  .  .  ,  X„  are  the  values  of  X  for 


X  =  Xi, 

.  .  .  f  X  —  2^nj  set 

Sfc  =  ^xi\     Sk  = 

SXi^ 

Then 

Si  =  niiQ  +  UiSi  +  M2S2  +  •  • 

•     +  Un-lSn-X, 

(24) 

S2   =  WW20  +  U21S1  +  1*22-52  +     • 

•     +  U2  n-lSn-h 

Sn  =  nUnd  +  UnlSl  +  W„2S2  + 

•    '    •     +  Itn  n-lSn-l' 

80  THEORY  OF  EQUATIONS  tCn.  vil 

Since  the  elementary  symmetric  functions  of  Xi,  .  .  .  ,  Xn  arc  expres- 
sible in  terms  of  >Si,  *S2,  .  .  .  ,  ^S^  (§  6),  we  can  find  the  coefficients  of  the 
equation  having  the  roots  Xi,  .  .  .  ,  Xn'. 

(25)  X"  +  PiX«-i  +  P2X"-2  +  .  .  .  +  P„  =  0. 

Another  method  of  forming  this  equation  is  given  in  Ch.  XII,  §  9. 

If  we  seek  values  of  Uo,  ih,  .  .  .  ,  w„_i,  such  that  Pi,  P2,  .  .  .  ,  P*  shall 
all  vanish  and  therefore  Si  =  S2  =  •  •  •  =  Sk  =  0,  by  Newton's  identities 
(7),  we  have  only  to  satisfy  a  system  of  k  equations  [see  (24)]  homogeneous 
in  «o,  .  .  .  ,  Un-i  and  of  degrees  1,  2,  .  .  .  ,  k,  respectively.  In  partic- 
ular, Si  =  0  enables  us  to  express  Uo  in  terms  of  Ui,  .  .  .  ,  so  that 


(26) 


x=..(.-^)+4-l)+ •••+.»-.(-— ^> 


Example.     For  n  =  3,  X  =  ui(x  —  I  si)  +  uoix-  —  ^  §2), 

S2  =  2X1^  -  Ui%S2  -  i  Si^)  +  2  UMSZ  -   I  S1S2)  +  U^KSi  -  i  82^). 

Thus  S2  =  0  gives  

(3  S2  —  sr)  ui  =  (siSo  —  3  S3  +  V  — 3  A)«2, 

A  =  S0S2S4  +  2  S1S2S3  —  S0S3"  —  81^84  —  S2^ 

Hence  the  cubic  equation  is  reduced  to  X''  +  P3  =  0  by  the  substitution 

X  =  (.si.so  -  3  .S3  +  V-3A) (3  X  -  si)  +  (3  .S2  -  .s'i-)(3  x^  -  S2). 

By  Ex.  6,  p.  158,  A  is  the  discriminant  of  the  cubic  equation. 

EXERCISES 

l.f  For  71  =  4,  take  2^3  =  0  in  (26)  and  find  the  cubic  equation  for  Ui/1/2  which 
results  from  P3  =  0  (i.e.,  *S3  =  0,  since  *Si  =  0).  The  new  quartic  equation 
X^  +  F2X'  +  P4  =  0  may  be  solved  in  terms  of  square  roots. 

2.t  For  n  =  5,  the  condition  for  »S2  =  0  is  that  a  certain  quadratic  form  q  in 
Ui,  .  .  .  ,  Ui  shall  vanish.  Now  q  can  l)e  expressed  as  a  sum  of  the  squares  of 
foiu-  linear  functions  Ly  of  Ui,  .  .  .  ,  u-i.  Taking  Li  =  1X2,  L3  =  iLi,  where 
{2  =  —  1^  we  have  Si  =  0.  By  means  of  tlie  r(!sulling  two  linear  relations  between 
Ui,  .  .  .  ,  W4,  we  may  express  S3  as  a  cubic  function  of  «i,  112,  for  example.  We 
must  therefore  solve  a  cubic  equation  in  Ui/mo  to  find  the  u's  making  also  S3  =  0. 
Tlie  new  quintic  equation  is  X^  +  PiX  +  P5  =  0.  If  Pi  ^  0,  set  X  =  ?/  \^i. 
Then  ?/  +  ?/  +  c  =  0.     (Bring,  1786;  Jcrrard,  1834.) 


CHAPTER  VIII 

Reciprocal  Equations.    Construction  of  Regular  Polygons. 
Trisection  of  an  Angle 

1.  For  certain  types  of  equations,  such  as  reciprocal  and  binomial 
equations,  there  exist  simple  relations  between  the  roots,  and  these  relations 
materially  simplify  the  discussion  of  the  equations. 

An  equation  is  called,  a  reciprocal  equation  if  the  reciprocal  of  each  root 
is  also  a  root.  Apart  from  possible  roots  1  and  —1,  each  of  which  is  its 
own  reciprocal,  the  roots  are  in  pairs  reciprocals  of  each  other. 

For  example,  the  equation 

fix)  =  (x-l)(x2-ix  +  l)  =  0 

is  a  reciprocal  equation  having  the  roots  1,2,  5.  If  we  replace  x  by  1/a;  and  multi- 
ply the  resulting  function  by  x^,  we  get  —f{x).  Here  (1)  holds  for  n  =  3  and  for 
the  minus  sign. 

In  general,  if 

fix)  ^x^-i-  '  •  ■  +c  =  0 

is  a  reciprocal  equation,  no  root  is  zero,  so  that  c  5^  0.  If  r  is  any  root  of 
f{x)  =  0,  1/r  is  a  root  of  f{l/x)  =  0,  and  hence  of 


../(i).i  + 


+  ex""  =  0. 


Since  the  former  is  a  reciprocal  equation,  it  has  the  root  1/r.     Hence 
any  root  of  the  former  equation  is  a  root  of  the  new  equation.     Thus,  by 

(1)  and  (2)  of  Ch.  VI,  the  left  member  of  the  latter  is  the  product  of  f{x) 
by  c.     Then,  by  the  constant  terms,  1  =  C".     Hence  c  =  ±1  and 

(1)  x"/Q  =  ±/(x). 

Thus  if  p,-  a:"""*  is  a  term  of /(x),  also  ±  p,.^'  is  a  term.     Hence 

(2)  f{x)  =  x"  ±  1  +  pi(a:"-i  _t.  _^)  _|_  p2(x"-2  ±  x^)  +  •  •  •  . 

81 


82  THEORY  OF  EQUATIONS  [Ch.  vill 

If  n  is  odd,  n  =  2  t  -\-  1,  the  final  term  is 

and  a:  ±  1  is  a  factor  oif{x).     In  view  of  (1),  the  quotient 
has  the  property  that 

Hence  Q{x)  =  0  is  a  reciprocal  equation  of  the  type 

(3)  a;2'  +  1  +  ci(.x2'-i  +  x)  +  c^ix'-'-'-  +  a;^)  +  •  •  •  +  Ctx'  =  0. 

Indeed,  the  highest  power  x-'  of  a:  has  the  coefficient  unity  and  the  con- 
stant term  is  unity,  so  that  it  is  of  the  form  (2)  with  the  upper  signs. 

If  n  is  even,  n  =  2t,  and  if  the  upper  sign  holds  in  (1),  we  have  just  seen 
that  (2)  is  of  the  form  (3).  Next,  let  the  lower  sign  hold  in  (1).  Then 
Pt  =  0,  since  a  term  ptX^  would  imply  a  term  —pix^.  The  final  term  in 
(2)  is  therefore 

Hence /(x)  has  the  factor  x-  —  \.  As  before,  the  quotient  is  of  the  form  (3). 
In  each  case  we  have  been  led  to  a  reciprocal  equation  of  type  (3). 
The  solution  of  the  latter  may  be  reduced  to  the  solution  of  an  equation  of 
degree  t  and  certain  quadratic  equations.  To  prove  this,  divide  the  terms 
of  (3)  by  xK     Then 

(4)  (,.+^)  +  ,,(,.-.  +  _L^)  +  ,,(,.-.  +  _^ 

+  •  •  •  +c,_i(.r  +  ^)  +  c,  =0. 

To  reduce  this  to  an  equation  of  degree  t,  we  set 

(o)  X  +  -  =  2. 

Then 

a;2  +  -,  =  22  _  2,     x^^\  =  z"  -?>z,  .  .  .  , 
X-  .r' 

while  the  general  binomial  in  (4)  can  be  computed  from 
(6)  ^  +  S  =  K"""+?-)-(^"'+^} 


§  11  RECIPROCAL  EQUATIONS  83 

For  example, 

a;4  +  1  =  2(^3  _  3  2)  _  (^2  _  2)  =  ^4  _  4  22  -I-  2. 

However,  we  can  obtain  an  explicit  expression  for  x''  +  l/x''  by  noting 
that  it  is  the  sum  of  the  A;th  powers  of  the  roots  x,  1/x  of 

2/'  -  (x  +  ~jy  +  X  .-  =  ?/-  2?/  +  1  =  0. 

The  sum  of  the  kth  powers  of  the  roots  oi  if  —  py  -^  q  —  0  was  found 
in  Ex.  2,  p.  75.     Taking  p  =  z,  q  =  I,  wc  have 

/^x    :    ,    1         ,       1  ,  0   ,   Hk-3)   ,  ,      k(k  -  4)(fc  -  5)   ,  ,  , 

(7)  a;*  +  ^  =z'  -  kz'^-'-  +    \^2     ^      ~         1.2.3 ^      +  '  "  ' 

fc(fe-r-l)(fc-r-2;  .  .  .  (fc-2r+l)  ^  ^ 

+  ^     ^^  1  .  2 . 3  .  .  .  r  ^        ^ 

Hence  (4)  becomes  an  equation  of  degree  t  in  z.  From  each  root  z  we 
obtain  two  roots  x  of  (3),  which  are  reciprocals  of  each  other,  by  solving 
the  quadratic  equation  x-  —  zx  -{-  1  =  0,  equivalent  to  (5). 

Example.  Solve  x^  —  5  x*  -\-  9  x^  —  9  x"^  -{-  5  x  —  1  =  0.  Dividing  by  a;  —  1, 
we  get  X*  -  4  x^  +  5  .c2  -  4  X-  +  1  =  0.     Thus 

x2  +  -,  -  4  (x  +  -  )  +  5  =  0,     z~-4z  +  3  =  0,     z  =  1ot3. 
x^  \        x/ 

For  2  =  1,  x2  -  a;  +  1  =  0,  x  =  ^  (l  ±  V^).  For  2  =  3,  x^  -  3x  +  1  =  0, 
X  =  §  (3  ±  V5) .    These  with  x  =  1  give  the  five  roots. 

EXERCISES 

Solve  by  radicals  the  reciprocal  cciuations 

1.   x^  -  7  x"  +  x3  -  x2  +  7  X  -  1  =  0.  2.   x^  =  1. 

3.  x«  =  1.  4.  x^  +  1  =  0. 

5.  Find  the  2;-cubic  for  x''  =  1. 

6.  Find  the  s-quintic  for  x^^  =  1. 

7.  The  2-quartic  for  x^  =  1  is  3"  +  2^  -  3  2^  -  2  2  +  1  =  0.  It  has  the  root  - 1 
since  the  2-equation  for  x^  =  1  is  2  +  1  =  0.  Verify  that,  on  removing  the  factor 
2+1  from  the  quartic,  we  get  the  2-cubic  2^  —  3  2  +  1  =  0  for  (x^  —  1) /(x^  —  1)  =  0. 

8.  What  are  the  trigonometric  representations  of  the  roots  of  the  2-equations  in 
Exs.  5  and  6?     Hint:    if  x  =  cos  0  +  i  sin  e,  1/x  =  cos  d  —  i  sin  e. 


84  T I! FA) RY   OF  EQUATIONS  [Ch.  VIII 

2.  Binomial  Reciprocal  Equations.  A  reciprocal  equation  with  only 
two  terms  is  of  the  form  x"  ±  1  =  0.  Its  roots  were  expressed  in  terms 
of  trigonometric  functions  in  Ch.  II.  But  now  we  wish  to  use  only  alge- 
braic methods.*  We  might  proceed  as  in  §  1,  first  **  removing  the  factor 
a:  ±  1  (if  n  is  odd)  or  a;^  —  1  (if  n  is  even  and  the  lower  sign  holds),  and 
then  applying  substitution  (5)  to  obtain  the  ^-equation.  Except  for  special 
values  of  n  (as  those  in  Exs.  2-6,  §  1),  there  is  a  more  effective  method, 
leading  to  auxiliary  equations  of  lower  degree  than  the  2-equation.  For  in- 
stance, it  will  be  shown  that  x"  —  1  =0  can  be  solved  in  terms  of  square 
roots;  it  is  only  a  waste  of  effort  to  form  the  z-equation  of  degree  8. 

3.  The  new  method  will  first  be  illustrated  for  x'  —  \  =  ^  since  it  then 
differs  only  in  form  from  the  earlier  method  of  treating  reciprocal  equations. 
Removing  the  factor  x  —  1,  we  have 

(8)  x^-\-x^-\-x^  +  x''  +  x'-\-x  +  l  =  0. 

If  r  is  a  particular  root  of  (8),  its  six  roots  are  (Ch.  II,  §  13), 

(9)  r,  r-,  r\  r\  r\  r\ 

By  the  substitution  (5),  we  obtain  the  cubic  equation 

(10)  2^  _[_  ^2  _  2  2  -  1  =  0, 
whose  roots  are  therefore 

(11)  zi  =  r  +  -  =  r  -{-  r^,     Zi^  r^  +  -  =  f-  +  r^     23  =  r^  +  -^  =  r^  +  ?'■*• 

r  r-  7"* 

The  new  method  consists  in  starting  with  these  sums  of  pairs  of  the 
six  roots  and  forming  the  cubic  equation  having  these  sums  as  its  roots. 
Since  r  is  a  root  of  (8), 

22i  =  r  +  r-  +•••+?•«=- 1,     ^z^zi  =  2  (r  +  •  •  •  +  r*')  =  -2, 

2i2o23  =  2  +  r+  •  •  •   +r'^=  1. 

Hence  Z\,  Zo,  zz  are  the  roots  of  (10).     If  a  root  Z\  be  found,  we  can  obtain 
r  from  the  quadratic  equation  r^  —  z^r  -\-  \  =  0. 

*  It  is  an  important  fact,  not  proved  or  used  hevo,  that  .c"  rfc  1  =  0  is  solvable  by 
radicals,  namely,  by  a  finite  number  of  applications  of  the  operation  extraction  of  a 
single  root  of  a  known  number.  Cf.  Dickson,  I ntroduction  to  the  Theory  of  Algebraic 
Equations,  John  Wiley  &  Sons,  pp.  77,  7S.  Note  that  it  suffices  to  treat  the  case  n 
prime,  since  .r''^=  .1  is  equivalent  to  the  chain  of  equations  y''  =  A,  x^  =  y. 

**  If  n  =  pq,  we  may  remove  the  factors  x^  ±  1  if  p  is  odd.     See  Ex.  7,  §1. 


§4.51  RECIPROCAL  EQUATIONS  85 

We  can,  however,  find  r  by  solving  first  a  quadratic  equation  and  after- 
wards a  cubic  equation.     To  this  end,  set 

(12)  Ui  ^  r  -\-  r"  +  r*,  y-i  ^  r^  -{-  r^  -\-  f^. 

Then 

2/1  +  ?/2  =  - 1,  yiV'i  =  3  +  r+---+r«  =  2, 

so  that  i/i  and  y^  are  the  roots  of 

2/2  +  2/  +  2  =  0. 

Then  r,  r~,  r*  are  seen  to  be  the  roots  of 

p3  -  2/ip-  +  yop  -1  =  0. 

4.t  The  Periods.  We  now  explain  the  principle  discovered  by  Gauss 
by  which  we  select  the  pairs  from  (9)  to  form  the  periods  Zi,  Z2,  Zs  in  (11), 
and  the  triples  to  form  the  periods  yi,  y^  in  (12).  To  this  end  we  seek  an 
integer  g  such  that  the  six  roots  (9)  can  be  arranged  in  the  order 

(13)  r,  r»,  ro\  r<}\  /■'/',  r»'\ 

each  term  being  the  ^th  power  of  its  predecessor.  The  choice  ^  =  2  is 
not  permissible,  since  the  fourth  term  would  then  be  r^  =  r.  But  we  may 
take  g  =  S,  and  the  desired  order  is 

(14)  r,  r^,  r^,  r^,  r^,  r^, 

each  term  being  the  cube  of  its  predecessor.  To  form  the  two  periods 
yi  and  7/2,  each  of  three  terms,  we  take  alternate  terms  of  (14).  To  form 
the  three  periods  Zi,  z-y,  23,  each  of  two  terms,  we  take  any  one  of  the  first 
three  terms  (as  r^)  and  the  third  term  after  it  (then  r^). 

5.t  Solution  of  x^'^  =  1  by  Square  Roots.  Let  r  be  a  root  9^  1. 
Then 

r"  —  1 

[ ±  _   ,,16  4-  ^15  _^     .    .    .     ^  ^  4_  1   ^  0. 

r  —  1 

As  in  §  4,  we  may  take  gr  =  3  and  arrange  the  roots,  r,  .  .  .  ,  r^^  so  that 
each  is  the  cube  of  its  predecessor: 

M        mO        y*V       /V»1U       ,y»lo        ^5       ^15        <^  1 1        -y»lb        ,y<14        y%^       lyi         ^4        yt\Z        ^2        <v*D 

Taking  alternate  terms,  we  form  the  2  periods  each  of  8  terms: 

7/1  =  r  +  7-9  +  ri'  +  ri5  +  r^^  +  r»  +  r"  +  r"^, 
1/2  =  r^'  +  r"'  +  r^  +  r"  +  P*  +  r'^  +  r'^  +  r^ 


86  THEORY  OF  EQUATIONS  [Ch.  vill 

Hence  yi  +  y2=  —1.     We  find  that  ?/i?/2  =  4  (r -|-  •  •  •  -\-r^^)=  —4.     Thus 

(15)  yi,  2/2     satisfy     7/  -^  y  -  -i  =  0. 
Taking  alternate  terms  in  yi,  we  form  the  two  periods 

Zi  =  r  -{-  r'^  +  r^^  +  r*,     2-2  =  r^  -{-  r^'"  +  r^  +  r^. 
Taking  alternate  terms  in  y2,  we  form  the  two  periods 

Wi  =  r^  +  r^  +  r^''  +  r^-,     w-z  =  r^"  +  r"  +  r^  +  r^. 

Thus  Zi-{-  Z2  =  yi,  Wi  -\-  w-i  =  y-y.     We  find  that  2122  =  Wiw-z  =  —  1. 
Hence 

(16)  Zi,  Zo     satisfy       z-  —  yiZ  —1=0, 

(17)  1^1,  lOo     satisfy      10-  —  y^w  —1  =  0. 
Taking  alternate  terms  in  Zi,  we  have  the  periods 

Vi  =  r  +  r^^,     Vo  =  r^^  -\-  r*. 
Now,  Vi  +  t'2  =  Zi,  yiy2  =  Wi.     Hence 

(18)  I'l,  Vo     satisfy     v-  —  Ziv  +u'i  =  0, 

(19)  r,  ri«     satisfy     p-  -  Vip  +1=0. 

Hence  we  can  find  r  by  solving  a  series  of  quadratic  equations.  Which 
of  the  sixteen  values  of  r  we  shall  thus  obtain  depends  upon  which  root 
of  (15)  is  called  2/1  and  which  7/2,  and  similarly  in  (16)-(19).  We  shall  now 
show  what  choice  is  to  be  made  in  each  such  case  in  order  that  we  shall 
finally  get  the  value  of  the  particular  root 

2  7r   ,    .   .    2x 
r  =  cos  "YY  +  I  sm  — • 

Then 

1  27r       .   .    2x  ,1       „        27r 

-  =  cos  ^  -  i  sm  — ,      ri  =  r  +  -  =  2  cos  — , 

.  O  TT     ,       .     .       8  TT  ,      ,       1  ^  8  TT 

r^  =  COS  ~-  +  I  sm  -^>       v-z  =  r'*  +  — ,  =  2  cos  —=■• 
1/  1/  r*  17 

Hence  Vi  >  V'>  >  0,  and  therefore  Zi  >  0.     Similarly, 

,,1,,,1       _,        6x,_        10  TT      „        Gtt      ^        7t^„ 
t^i  =  r^  +  -  4-  r^  +  -.  =  2  cos  v^  +  2  cos  — -=-  =  2  cos  v=r  —  2  cos  — r  >  0, 
r^  r^  17  17  17  17 

„         Gtt    ,    -         IOtt    ,    ^         12  7r    ,    _         14  7r   ^,, 
2/2  =  2cos— +  2cos-Yy-  +  2cos— y-  +  2cos--y-<  0, 


i  6] 


CONSTRUCTION  OF  REGULAR  POLYGONS 


87 


since  only  the  first  cosine  in  y-y  is  positive  and  it  is  numerically  less  than 
the  third.     But  ijiijo  =  -4.     Hence  t/i  >  0.     Thus  (15)-(17)  give 

2/2  =  ^-^17-1), 


yi  =  \  Wvi  -  1), 


^\  =  \  yi  +  Vl  +  i7/r, 

We  now  have  the  coefficients  of  (18)  and  know  that  Wi  >  ^J2  >  0.  These 
results  are  sufficient  for  the  next  problem.  Of  course,  we  could  go  on 
and  obtain  the  explicit  expression  for  Vi,  and  that  for  r  in  terms  of  square 
roots. 

6.t  Construction  of  a  Regular  Polygon  of  17  Sides.  In  a  circle  of 
radius  unity,  construct  two  perpendicular  diameters  AB,  CD,  and  draw 


tangents  at  A,  D,  which  intersect  at  S  (Fig.  20).     Find  the  point  E  in 
AS  for  which  AE  =  I  AS,  by  means  of  two  bisections.     Then 

AE  ^l,     OE  =  \  Vl7. 

Let  the  circle  with  center  E  and  radius  OE  cut  AS  at  F  and  F'.    Then 

AF  =^EF  -EA=OE-\  =  l  y„ 

AF'  =  EF'  -{-EA  =0E  +  \=  -^y^, 

OF  =  VOA'  +  AF'  =  Vl  +  1  y{',     OF'  =  VTTIy?- 

Let  the  circle  with  center  F  and  radius  FO  cut  AS  at  H,  outside  of  F'F; 
that  with  center  F'  and  radius  F'O  cut  AS  at  H'  between  F'  and  F.     Then 

AH  =  AF  +  FH  =  AF  +  OF  =  \y,  +  VTThA'  =  ^i, 
AH'=  F'H'  -  F'A  =  OF'  -  AF'  =  wi. 


88 


THEORY  OF  EQUATIONS 


[Ch.  viri 


It  remains  to  construct  the  roots  of  equation  (18).  This  will  be  clone 
as  in  Ch.  I,  §  16.  Draw  HTQ  parallel  to  AO  and  intersecting  OC  pro- 
duced at  T.  Make  TQ  =  AH'.  Draw  a  circle  having  as  diameter  the 
line  BQ  joining  B  =  (0,  1)  with  Q  =  (21,  lOi).  The  abscissas  ON  and  OM 
of  the  intersections  of  this  circle  with  the  a;-axis  OT  are  the  roots  of  (18). 
Hence  the  larger  root  Vi  is  OM  =  2  cos  2  7r/17. 

Let  the  perpendicular  bisector  LP  of  OM  cut  the  initial  circle  of  unit 
radius  at  P.     Then 

2t 


cos  LOP  —  OL  —  cos 


17 


LOP^^- 


Hence  the  chord  CP  is  a  side  of  the  inscribed  regular  polygon  of  17  sides, 
constructed  with  ruler  and  compasses. 


1.   For  71  =  5, 


EXERCISES 

2,  the  periods  are  r  +  r^,  r-  +  r^.  Show  that  they  are  the 
roots  of  the  z-quadratic  obtained  in  Ex.  2,  p.  83. 

2.t  For  n  =  13,  find  the  least  g,  form  the  three 
periods  each  of  four  terms,  and  find  the  cubic  havhig 
them  as  roots. 

3.  For  ?(  =  5,  E.\.  1  gives  r  +  /■■*  =  2  cos  2  tt/o  = 
5  ( V  5  —  1).  In  a  circle  of  radius  unity  and  center  0 
draw  two  perpendicular  diameters  ADA',  BOB'. 
With  the  middle  point  .1/  of  OA'  as  center  and  radius 
MB  draw  a  circle  cutting  OA  at  C  (Fig.  21).  Show 
that  OC  and  BC  are  the  sides  sio  and  ss  of  tlie 
inscribed  regular  decagon  and  pentagon  rcspectivel3\ 
Hints: 

MB  -  I V5,   OC  =  I  {VE -  1),   BC  =  Vi  +  OC-  =  h  ^/lo  -2V5, 

Sio  =  2  sin  18°  =  2  cos  ^  =  OC, 
o 

si"  =  (2  sin  3G°)2  =  2  [1  -  cos^-j  =  i  (lO  -  2  V^),     s,  =  BC. 


7.  t  Regular  Polygon  of  n  Sides.  If  n  be  a  prime  such  that  n  —  1  is 
a  power  2''  of  2  (as  is  the  case  when  n  =  3,  5,  17),  the  n  —  1  imaginary  nth 
roots  of  unity  can  be  separated  into  2  sets  each  of  2''~^  roots,  each  of  these 
sets  subdivided  into  2  sets  each  of  2''~-  roots,  etc.,  until  we  reach  the 


8  81  CONSTRUCTION  OF  REGULAR  POLYGONS  89 

sets  r,  1/r  and  r-,  l/r-,  etc.,  and  in  fact  *  in  such  a  manner  that  we  have  a 
series  of  quadratic  equations,  the  coefficients  of  any  one  of  which  depend 
only  upon  the  roots  of  quadratic  equations  preceding  it  in  the  series. 
Note  that  this  was  the  case  for  n  =  17  (§  5)  and  for  n  =  5.  It  is  in  this 
manner  that  it  can  be  proved  that  the  roots  of  x"  =  1  can  be  found  in 
terms  of  square  roots,  so  that  a  regular  polygon  of  n  sides  can  be  inscribed 
by  ruler  and  compasses,  provided  n  be  a  prime  of  the  form  2''  +  1. 

If  n  be  a  product  of  distinct  primes  of  this  form,  or  2^  times  such  a  prod- 
uct (for  example,  n  —  15,  30  or  6),  or  if  n  =  2'"  (w  >  1),  it  follows  readily 
that  we  can  inscribe  by  ruler  and  compasses  a  regular  polygon  of  n  sides. 
But  this  is  impossible  for  other  values  of  7i.  This  impossibility  will  be 
proved  for  n  =  7  and  n  =  9,  the  method  of  proof  being  applicable  to  the 
general  case. 

8.  Regular  Polygons  of  7  and  9  Sides;  Trisection  of  an  Angle.     For 

brevity  we  shall  occasionally  use  the  term  "  construct  "  for  "  construct 
by  ruler  and  compasses."  If  it  were  possible  to  construct  a  regular  poly- 
gon of  7  sides  and  hence  angle  2  7r/7,  we  could  construct  a  line  of  length 
2  cos  2  7r/7,  the  base  of  a  right-angled  triangle  whose  hypotenuse  is  of 
length  2  and  one  of  whose  acute  angles  is  2  t/7.     Set 


Then 


27r   ,    .    .    27r 
r  =  cos-=-  -f- 1  sm  -=-• 


1  27r        .   .    27r  ,    1     ^o        27r 

-  =  cos  — I  sm  _-  >    r  -\-  -  =  2  cos  -^  • 

r  ^7  7      ^        r  7 


Hence  2  cos  2  7r/7  is  a  root  of  the  cubic  equation  (10).  This  equation  has 
no  rational  root.  For,  if  it  had  a  rational  root,  it  would  have  (Ch.  VI, 
§8,  §5)  an  integral  root  which  is  a  divisor  of  the  constant  term  —  1, 
whereas  neither  +  1  nor  —  1  is  a  root.  Hence  we  shall  know  that  it  is  im- 
possible to  construct  a  regular  polygon  of  7  sides  by  ruler  and  compasses 
as  soon  as  we  have  proved  (§10)  the  next  theorem. 

*  See  the  author's  article  "Constructions  with  ruler  and  compasses;  regular  poly- 
gons," in  Monographs  on  Topics  of  Modern  Mathematics,  edited  by  J.  W.  A.  Young, 
Longini^ns,  Green  and  Co.,  New  York,  1911,  p.  374.  In  addition  to  the  references  there 
given  (p.  386),  mention  should  be  made  of  the  book  by  Klein,  Elementarmathematik  vom 
Hoheren  Standpunkle  aus,  Leipzig,  1908,  vol.  1,  p.  125;  and  ed.  2,  1911. 


90  THEORY  OF  EQUATIONS  ICh.  vm 

Theorem.  It  is  not  possible  to  construct  by  ruler  and  compasses  a  line 
whose  length  is  a  root  of  a  cubic  equation  with  rational  coefficients  but  having 
no  rational  root. 

This  theorem  shows  also  that  it  is  not  possible  to  construct  a  regular 
polygon  of  9  sides  and  hence  that  it  is  not  possible  to  construct  the  angle 
40°  by  ruler  and  compasses.  Indeed,  if  r  =  cos  40°  +  i  sin  40°,  then 
r  +  1/r  =  2  cos  40°  is  a  root  (Ex.  7,  p.  83)  of 

2^  -  3  2  +  1  =  0. 

The  same  equation  follows  also  from  the  identity 

cos  3  A  =  4  cos^  A  —  3  cos  A 

by  taking  A  =  40°,  replacing  cos  120°  by  its  value  —  ^,  and  setting 
2  =  2  cos  40°.  Since  neither  divisor  1  nor  —  1  of  the  constant  term  is  a 
root  of  the  2-cubic,  there  is  no  rational  root. 

Corollary.     It  is  not  possible  to  trisect  every  angle  by  ruler  and  compasses. 

Indeed,  angle  40°  cannot  be  constructed,  while  angle  120°  can  be. 

9.  Duplication  of  a  Cube.  Another  famous  problem  of  antiquity  was 
the  construction  of  a  cube  whose  volume  shall  be  double  that  of  a  given 
cube.  Take  the  edge  of  the  given  cube  as  the  unit  of  length  and  denote 
by  X  the  length  of  an  edge  of  the  required  cube.  Then  a;^  —  2  =  0. 
Since  no  one  of  the  divisors  of  2  is  a  root  of  this  cubic  equation,  the  theorem 
stated  in  §  8  implies  the  impossibility  of  the  duplication  of  a  cube  by 
ruler  and  compasses. 

10.  t  Cubic  Equations  with  a  Constructible  Root.  It  remains  to  prove 
the  theorem  in  §  8  from  which  we  have  drawn  such  important  conclusions. 
Suppose  that 

(20)  x^  +  a.T2  -^  j3x-\-y  =  0  (a,  /3,  y  rational) 

is  a  cubic  equation  having  a  root  Xi  such  that  a  line  of  length  Xi  or  —  .Ti 
can  be  constructed  by  ruler  and  compasses.  We  shall  prove  that  one  of 
the  roots  of  (20)  is  rational. 

The  construction  is  in  effect  the  determination  of  various  points  as  the 
intersections  of  auxiliary  straight  lines  and  circles.  Choose  rectangular 
axes  of  coordinates.  The  coordinates  of  the  intersection  of  two  straight 
lines  are  rational  functions  of  the  coefficients  of  the  equations  of  the  two 


§  10)  GEOMETRICAL  CONSTRUCTIONS  91 

lines.  To  obtain  the  coordinates  of  the  intersection  of  the  straight  Une 
y  =  mx  +  b  with  the  circle 

(.T  -  pY  +  (y  -  q)-  =  r2, 

we  eliminate  y  and  obtain  a  quadratic  equation  for  x.  Thus  x,  and  hence 
also  y,  involves  no  irrationality  (besides  irrationalities  in  ni,  b,  p,  q,  r)  other 
than  a  square  root.  Finally,  the  intersections  of  two  circles  are  given 
by  the  intersections  of  one  of  them  with  their  common  chord,  so  that  this 
case  reduces  to  the  preceding.  Hence  the  coordinates  of  the  various  points 
located  by  the  construction,  and  therefore  also  the  length  ±  Xi  of  the  seg- 
ment joining  two  of  them,  are  found  by  a  finite  number  of  rational 
operations  and  extractions  of  real  square  roots,  performed  upon  rational 
numbers  and  numbers  obtained  by  earlier  ones  of  these  operations. 

If  Xi  is  rational,  (20)  has  a  rational  root  as  desired.  Henceforth,  let  Xi 
be  irrational.  Then  .Xi  is  the  quotient  of  two  sums  of  terms,  each  term 
being  a  rational  number  or  a  rational  multiple  of  a  square  root.  A  term 
may  involve  superimposed  radicals  as 

r  =  VlO  -  2\/5,     s  =  VlO  +  2  V5,     ^  =  \/4  -  2  V3. 

But  t  equals  Vs  —  1  and  would  be  replaced  by  that  simpler  value.  As  a 
matter  of  fact,  r  is  not  expressible  rationally  *  in  terms  of  a  finite  number 
of  square  roots  of  rational  numbers,  and  is  said  to  be  a  radical  of 
order  2.  A  term  having  n  superimposed  radicals  is  of  order  n  if  it  is  not 
expressible  rationally  in  terms  of  radicals  each  with  fewer  than  n  super- 
imposed radicals.  In  case  a:i  =  2  r  —  7  s,  we  would  express  Xi  in  the  form 
2  r  —  28  Vs/r,  involving  a  single  radical  of  order  2;  indeed,  rs  =  4  Vs. 
If  Xi  involves  Vs,  Vs  and  Vlo,  we  replace  Vl5  by  Vs  •  Vs. 

We  may  therefore  assume  that  no  one  of  the  radicals  of  highest  order 
n  in  Xi  is  a  rational  function  with  rational  coefficients  of  the  remaining 
radicals  of  order  n  and  radicals  of  lower  order,  that  no  one  of  the  radicals 
of  order  n  —  1  is  a  rational  function  of  the  remaining  radicals  of  order 
n  —  \  and  radicals  of  lower  order,  etc. 

Let  Vk  be  a  radical  of  highest  order  n  in  a;i.     Then 

a  +  bVk 

Xi  =  7=> 

c-^dVk 
*  That  is,  as  a  rational  integral  function  with  rational  coefficients. 


92  THEORY  OF  EQUATIONS  [Ch.  viil 

where  a,  .  .  .  ,  d  do  not  involve  Vk,  but  may  involve  other  radicals  of 
order  n.  U  d  9^  0,  Vk  9^  c/d,  in  view  of  the  preceding  assumption. 
Thus  we  may  multiply  the  numerator  and  denominator  oi  Xihy  c  —  d  Vk. 
Hence,  whether  d  9^  0  or  d  =  0,  we  have 

Xi  =  e+fVk  (f^O), 

where  neither  e  nor  /  involves  Vk.  Since  Xi  is  a  root  of  (20),  we  have 
A  -\-  B  Vk  =  0,  where  A  and  B  are  polynomials  in  e,  f,  k,  a,  /3,  7.  li  B  9^  0, 
we  could  express  Vk  as  a  rational  function  —A/B  of  the  remaining 
radicals  in  the  initial  Xi.  Hence  B  =  0  and  therefore  ^4.  =  0.  But  the 
result  of  substituting  e  —  f  Vk  for  x  in  the  cubic  function  (20)  is  evidently 
A  —  B  Vk.    Hence 

X2  =  e  —  f  Vk 

is  a  new  root  of  our  cubic  equation.     The  third  root  is 

X2  =  —a  —  Xi  —  Xo  =  —  a  —  2e. 

Now  a  is  rational.  If  e  is  rational,  X3  is  a  rational  root  of  (20),  as  desired. 
The  remaining  case  is  readily  excluded.  For,  if  e  is  irrational,  let  Vs  be 
one  of  the  radicals  of  highest  order  in  e.     Then,  as  above, 

X3  =  (j-\-hV~s  {h  9^  0), 

where  neither  g  nor  h  involves  Vs,  while  g  —  h  Vs  is  a  root  9^  Xs  of  (20), 
and  hence  identical  with  Xi  or  X2.     Thus 

e  ±  /  Vk  =  g  —  h  Vs. 

Now  Vs  and  all  the  radicals  appearing  in  g,  h,  s  occur  in  .T3  and  hence 
in  e.     But  Vk  is  not  expressible  in  terms  of  the  remaining  radicals  of  Xi. 
We  have  now  proved  that  if  the  constructible  root  .ri  of  (20)  is  irra- 
tional, there  is  a  rational  root  x^. 

11.  t  Problems  such  as  the  triscction  of  any  angle  can  often  bo  solved 
by  means  of  certain  curves.  We  note,  however,  that  there  exists  no  plane 
curve,  other  than  a  conic  section,  whose  intersections  b}^  an  arbitrary 
straight  line  can  be  found  by  ruler  and  compasses.* 

*  J.  Peterson,  Algtbraische  Gleichungen,  p.  169. 


CHAPTER  IX 

Isolation  of  the  Real  Roots  of  an  Equation  with  Real 

Coefficients 

1.  Method  of  RoUe.*  There  is  at  least  one  real  root  of  f'{x)  =  0  be- 
tween two  consecutive  real  roots  a  and  b  of  f(x)  =  0. 

For,  the  graph  oi  y  =  f{x)  has  a  bend  point  between  a  and  b. 

Corollary.  Between  two  consecutive  real  roots  r  and  s  of  fix)  =  0, 
lies  at  most  one  real  root  of  f{x)  =  0. 

For,  if  there  were  two  such  real  roots  a  and  b  of  the  latter  equation,  the 
first  theorem  shows  that/'(^)  =  0  would  have  a  real  root  between  a  and  b 
and  hence  between  r  and  s,  contrary  to  hypothesis. 

Now  f{x)  =  0  has  a  real  root  between  r  and  s  if  /(r)  and  f(s)  have  oppo- 
site signs  (Ch.  I,  §  12).         Hence  the  Corollary  gives  the 

Criterion.  If  r  and  s  are  consecutive  real  roots  off'{x)  =  0,  thenf{x)  =  0 
has  a  single  real  root  between  r  and  s  if  and  only  if  f{r)  and  f{s)  have  opposite 
signs.  At  most  one  real  root  of  f{x)  =  0  is  greater  than  the  greatest  real  root 
of  f'{.x)  =  0,  or  less  than  the  least  real  root  of  f'{x)  =  0. 

The  final  statement  follows  at  once  from  the  first  theorem. 

Example.    For  f{x)  =  3  x^  -  25  .c^  +  60  a;  -  20, 

tV/'(x)  =  x"  -  5  a;2  +  4  =  (.r^  -  l)(a;2  -  4). 
Hence  the  roots  of /'(.c)  =  0  are  ±1,  ±2.     Now 

/(_^)=_^,  /(_2)=-36,  /(-l)=-58,  /(1)  =  18,  /(2)=-4,  /(+^)=+^. 
Hence  there  is  a  single  real  root  in  each  of  the  intervals 
(-1,1),     (1,2),     (2,+^), 
and  two  imaginary  roots.     The  3  real  roots  are  positive. 

2.  The  first  theorem  of  §  1  is  a  special  case  of 

RoUe's  Theorem.  Between  two  consecutive  roots  a  and  b  of  f(x)  =  0, 
there  is  an  odA  number  of  real  roots  of  f'{x)  =  0,  a  root  of  multiplicity  m 
being  counted  as  m  roots. 

*  Traite  de  I'alyebrc,  Paris,  1G90.     Hudde  knew  the  method  in  1659. 

93 


94  THEORY  OF  EQUATIONS  [Ch.  ix 

We  may  argue  geometrically,  noting  that  there  is  an  odd  number  of 
bend  points  between  a  and  h,  the  abscissa  of  each  being  a  root  ol  J'{x)  =  0 
of  odd  multiplicity,  while  the  abscissa  of  an  inflexion  point  with  a  hori- 
zontal tangent  is  a  root  of  J'{x)  =  0  of  even  multiplicity. 

To  give  an  algebraic  proof,  let 

fix)  =  {x-  ay{x  -  hYQix),  a<h, 

where  Qix)  is  a  polynomial  divisible  by  neither  x  —  a  nor  x  —  h.     Then 

^ '^^^^  =  r{x  -  6)  +  .v(.r  -  a)  +  {x  -  a)(^  -  6) ^^. 

The  second  member  has  the  value  r{a  —  h)  <  0  ior  x  =  a  and  the  value 
s{b  —  a)  >  0  for  x  =  h,  and  hence  vanishes  an  odd  number  of  times  be- 
tween a  and  b  (Ch.  I,  §  12).  But,  in  the  left  member,  (x  —  a)(x  —  b)  and 
f{x)  remain  of  constant  sign  between  a  and  b,  since  f{x)  =  0  has  no  root 
between  a  and  b.     Hence /'(.r)  vanishes  an  odd  number  of  times. 

Corollary.  If  f{x)  =  0  has  only  real  roots,  f'{x)  =  0  has  only  real 
roots  distributed  as  follows:  an  (m  —  l)-fold  root  equal  to  each  m-io\d 
root  oi  f(x)  =  0  ioT  7n  ^  2;  a  single  root,  which  is  a  simple  root,  between 
two  consecutive  roots  of  /(.r)  =  0. 

For,  if  the  roots  of  /(.r)  =  0  are  a,  b,  c,  .  .  .  ,  arranged  in  ascending 
order,  of  multiplicities  r,  s,  t,  .  .  .  ,  respectively,  then  a,  b,  c,  .  .  .  are 
roots  of  f'{x)  =  0  of  multiplicities  r  —  1,  s  —  1,  t  —  1,  .  .  .  ,  and  between 
a  and  b  lies  at  least  one  real  root  of /'(.r)  =  0,  etc.  The  number  of  these 
roots  of /'(.r)  =  0  is  thus  at  least 

(r-l)  +  l  +  (.s-l)  +  l+(i-l)+  .  .  .  =r  +  s  +  t+  ■  ■  '  -1  =  11 -I, 
if  n  is  the  degree  of  /.  But  /'  is  of  degree  n  —  1  and  hence  has  only  these 
roots.     Thus  only  one  of  its  roots  lies  between  a  antl  b. 

EXERCISES 

1.  .r^  —  5  .c  +  2  =  0  has  1  negative,  2  positive  and  2  imaginary  roots. 

2.  x*  +  •?■  —  1  =  0  has  1  negative,  1  positive  and  -i  imaginary  roots. 

3.  x^  —  3  .r''  +  2  X-  —  5  =  0  has  two  iinaginarv  roots,  and  a  real  root  in  each 
of  the  intervals  (-2,  -1.5),  (-1.5,-  1),  (1,  2). 

4.  fix)  =  4  .T^  —  3  .r-»  -  2 .1-2  +  4  .r  -  10  =  0  has  a  single  real  root.     Hint: 

Fix)  =  I  fix)  =  5  .r"  -  3  .r^  -  .r  +  1  =  0 
has  no  real  root,  since  F'ix)  =  0  ha.s  a  single  real  root  and  for  it  F  is  positive. 

5.  If /(^"^X-r)  =  0  has  imaginary  roots,  fix)  =  0  has  imaginary  roots. 

0.  U  fix)  =  0  has  exactly  r  real  roots,  tlie  number  of  real  roots  of /(.r)  =  0  is 
r  +  1  or  is  less  than  r  +  1  by  an  even  number,  a  root  of  multiplicity  ?/i  being 
counted  as  m  roots. 


3] 


ISOLATION  OF  REAL  ROOTS 


95 


3.  Sturm's  Method.  Let  f(x)  =  0  be  the  given  equation  with  real 
coefficients,  and  f'{x)  the  first  derivative  of  f{x).  The  first  step  of  the 
usual  process  for  seeking  the  greatest  common  divisor  of  f{x)  and  f'(x) 
consists  in  dividing/  by/'  until  we  obtain  a  remainder  r{x),  whose  degree 
is  less  than  that  of  /'.  Then,  if  qi  is  the  quotient,  we  have  /  =  qj'  +  r. 
We  write /2  =  —r,  divide  /'  by/2,  and  denote  by/3  the  remainder  with  its 
sign  changed.     Thus 

/  =  (Zi/'  -  h    J'  =  (hh  -  h    h  =  q^f-s  -fi,.... 
The  latter  equations,  in  which  each  remainder  is  exhibited  as  the  nega- 
tive of  a  polynomial  fi,  yield  a  modified  process,  just  as  effective  as  the 
former  process,  for  finding  the  greatest  common  divisor  G  oi  f{x)  and/'(x) 
if  it  exists. 

Suppose  that  —fi  is  the  first  constant  remainder.  If  /»  =  0,  then  /a  =  G,  since 
/s  divides  fi  and  hence  also  /'  and  /  (by  using  our  equations  in  reverse  order) ;  while 
conversely,  any  common  divisor  of  /  and  /'  divides  /2  and  hence  also  fz. 

But  if  fi  is  a  constant  5^  0,  /  and  /'  have  no  common  divisor  involving  x.  This 
case  arises  if  and  only  ii  f{x)  =  0  has  no  multiple  root  (Ch.  I,  §  7),  and  is  the  only 
case  considered  in  §§4-6. 

Before  stating  Sturm's  theorem  in  general,  we  shall  state  it  for  a  numerical 
case  and  illustrate  its  use. 

Example.    f{x)  =  x^  +  4  x^  -  7.    Then  /'  =  3  .c^  +  8  x, 

f={\x+t)f'  -f„     /2  =  «^x  +  7, 

r  =  (Hx  +  fwv)/!-/!,  f^  =  m\. 

For  x  =  \,  the  signs  of  /,  /',  fo,  fz  are h  +  +,  showing  a  single  variation  of 

consecutive  signs.     For  x  =  2,  the  signs  are  +  +  +  +,  showing  no  variation  of 
signs.     Sturm's  theorem  states  that  there  is  a  single  real  root  between  1  and  2. 

For  X  =    —00,  the  signs  are 1 h,   showing  3   variations  of  signs.     The 

theorem  states  that  there  are  3  —  1  =  2  real  roots  between  —00  and  1.     Similarly, 


X 

Signs 

Variations 

- 1 

-2 
-3 
-4 

1    ++    1 
+  +    1      1 

111    + 
+  +  +  + 

1 

2 
2 
3 

Hence  there  is  a  single  real  root  between  —2  and  —1,  and  a  single  one  between 
—4  and  —3.  Each  real  root  has  now  been  isolated  since  we  have  found  two  num- 
bers such  that  a  single  real  root  lies  between  these  two  numbers  or  equals  one  of 
them. 


96  THEORY  OF  EQUATIONS  ICh.  ix 

4.  Sturm's  Theorem.  Let  f{x)  =  0  be  an  equation  with  real  coejficients 
and  without  multi'ple  roots.  Modify  the  usual  process  for  seeking  the  great- 
est common  divisor  of  f{x)  and  its  first  derivative*  fi(x)  by  exhibiting  each 
remainder  as  the  negative  of  a  'polynomial  fi: 

(1)  /  =   Qlfl   -  fh  /l   =   (hh  -  h  /2    =   ^3/3   -fi,...,    fn-2  =  Qn-lfn-l  "  /n, 

where  **  /„  is  a  constant  5^  0.  If  a  and  b  are  real  numbers,  a  <  b,  neither 
a  root  of  f{x)  =  0,  the  number  of  real  roots  of  f{x)  =  0  between  a  and  b  equals 
the  excess  of  the  number  of  variations  of  signs  of 

(2)  fix),  f,{x),  f,{x),  .  .  .  ,  fn-,(x),  /„ 

for  x  =  a  over  the  number  of  variations  of  signs  for  x  =  b.  Terms  which 
vanish  are  to  be  dropped  out  before  counting  the  variations  of  signs. 

For  brevity,  let  V^  denote  the  numl^er  of  variations  of  signs  of  the  num- 
bers (2)  when  a;  is  a  particular  real  number  not  a  root  of  /(.r)  =  0. 

First,  if  Xi  and  x^  are  real  numbers  such  that  no  one  of  the  continuous 
functions  (2)  vanishes  for  a  value  of  x  between  Xi  and  Xo  or  for  x  =  Xi  or 
X  =  X2,  the  values  of  any  one  of  these  functions  for  x  =  Xi  and  x  =  X2 
are  both  positive  or  both  negative  (Ch.  I,  §  12),  and  therefore  F^^  =  V^^. 

Second,  let  p  be  a  root  of  /,(.r)  =  0,  where  \  ^  i  <  n.     Then 

(3)  f^-.{x)  =5-/.(-r)-/.+i(.r) 

and  the  equations  (1)  following  this  one  show  that  fi-i(x)  and  fi(x)  have 
no  common  divisor  involving  x  (since  it  would  divide  the  constant  /„). 
By  hypothesis,  fi(x)  has  the  factor  x  —  p.  Hence  fi-i{x)  does  not  have 
this  factor  x  —  p.     Thus,  by  (3), 

/,_i(p)  =  -/.+i(p)  ^  0. 

Hence,  if  p  is  a  sufficiently  small  positive  number,  the  values  of 

fi-iix),     /.(.t),     fi+i(x) 

for  X  =  p  —  p  show  just  one  variation  of  signs,  since  the  first  and  third 
values  are  of  opposite  signs,  and  for  x  =  p  -\-  p  show  just  one  variation  of 

*  The  notation  /i  instead  of  the  usual  /',  and  similarly  /o  instead  of  /,  is  used  to  reg- 
ularize the  notation  of  all  the  /'s,  and  enables  us  to  write  any  one  of  the  equations  (1) 
in  the  single  notation  (3). 

**  If  the  division  process  did  not  yield  ultimately  a  constant  remainder  ?^  0,  /  and/i 
would  have  a  common  factor  involving  x,  and  hence  f{x)  =  0  a  multiple  root. 


§  41  ISOLATION  OF  REAL  ROOTS  97 

signs,  and  therefore  show  no  change  in  the  number  of  variations  of  sign 
for  the  two  values  of  x. 

It  follows  from  the  first  and  second  cases  that  F„  =  V^  ii  a  and  (3  are 
real  numbers  for  neither  of  which  any  one  of  the  functions  (2)  vanishes  and 
such  that  no  root  of  f(x)  =  0  lies  between  a  and  /3. 

Third,  let  r  be  a  root  of  f{x)  =  0.     By  Taylor's  Theorem  (8)  of  Ch.  I, 

Kr-p)  =  -vnr)  +  hvT{r)-   .  .  .  , 
fir  +  v)=      Pf'ir)  +  i  pT'ir)  +  .  .  .    . 

If  p  is  a  sufficiently  small  positive  number,  each  of  these  polynomials  in  p 
has  the  same  sign  as  its  first  term.  For,  after  removing  the  factor  p, 
we  obtain  a  quotient  of  the  form  ao  +  s,  where  s  =  aip  -\-  a2p^  +  .  .  . 
is  numerically  less  than  ao  for  all  values  of  p  sufficiently  small  (Ch.  I, 
end  of  §  11).     Hence  if /'(r)  is  positive,  /(r  —  p)  is  negative  and /(r  +  p) 

positive,  so  that  the  terms  f{x),  /i(x)  =  f'{x)  have  the  signs \-  for 

X  =  r  —  p  and  the  signs  +  +  for  x  =  r  +  p.     If  /'(r)  is  negative,  these 

signs  are  -\ and  —  —  respectively.     In  each  case,  f(x),  fi{x)  show  one 

more  variation  of  signs  ior  x  =  r  —  p  than  for  x  =  r  -{-  p.  Evidently  p 
may  be  chosen  so  small  that  no  one  of  the  functions  fi{x),  ...,/„  vanishes 
for  either  x  =  r  —  p  or  x  =  r  -\-  p,  and  such  that  fi{x)  does  not  vanish 
for  a  value  of  x  between  r  ~  p  and  r  -\-  p,  so  that  f{x)  =  0  has  the  single 
real  root  r  between  these  limits  (§1).  Hence  by  the  first  and  second* 
cases, /i,  .  .  .  ,  fn  show  the  same  number  of  variations  of  signs  ior  x  —  r  —  p 
and  X  =  r  -{-  p.     Thus,  for  the  entire  series  of  functions  (2),  we  have 

(4)  Vr-,  -    Vr+,   =    1. 

The  real  roots  of  f{x)  =  0  within  the  main  interval  from  a  to  6  (i.e.,  the 
aggregate  of  numbers  between  a  and  b)  separate  it  into  intervals.  By 
the  earlier  result,  Vx  has  the  same  value  for  all  numbers  in  the  same 
interval.     By  the  present  result  (4),  the  value  of  V^  in  any  interval  ex- 

*  The  argument  in  the  second  case  when  applied  for  i  =  I  requires  the  use  of 
/o  =  /  and  hence  does  not  indicate  the  variations  in  a  series  lacking  /.  To  avoid  the 
necessity  of  treating  this  case  i  =  1,  we  restricted  p  further  than  done  at  the  outset  so 
that  fi{x)  shall  not  vanish  between  r  —  p  and  r  +  P-  This  necessary  step  in  the  proof  is 
usually  overlooked.  Moreover,  we  have  not  adopted  the  usual  argument  based  upon 
the  continuous  change  of  x  from  a  to  b,  in  view  of  the  ambiguity  of  Vx  when  a;  is  a  root 
of /(x)  =  0,  etc. 


98  THEORY  OF  EQUATIONS  fCa.  IX 

ceeds  the  value  for  the  next  interval  by  unity.  Hence  Va  exceeds  Vb  by 
the  number  of  real  roots  between  a  and  b. 

Corollary.     If  a  <  6,     Va—  Vb. 

EXERCISES 

Isolate  by  Sturm's  theorem  the  real  roots  of 

1.   .T^  +  2  X  +  20  =  0.  2.   x^  +  X  -  3  =  0. 

5.  Simplifications  of  Sturm's  Functions.  In  order  to  avoid  fractions, 
we  may  first  multiply  /(.t)  by  a  positive  constant  before  dividing  it  by 
Ji{x),  and  similarly  multiply /i  by  a  positive  constant  before  dividing  it  by 
f2,  etc.  Moreover,  we  may  remove  from  any  Ji  any  factor  ki  which  is 
either  a  positive  constant  or  a  polynomial  in  x  positive  for  *  a  =  x  ^  h, 
before  we  use  that  fi  as  the  next  divisor. 

To  prove  that  Sturm's  theorem  remains  true  when  these  modified 
functions/,  Fi,  .  .  .  ,  Fm  are  employed  in  place  of  functions  (2),  consider 
the  equations  replacing  (1): 

fi  =  kyF,,     c,f=q,F,-hF,,     c,F,  =  q,F,-hF,, 

CiF-l   =   53^3-   kiFi,     .     .     .     ,    CmF,n-2   =   qm-\F m-l  —   k„,F,„, 

in  which  Cs,  Cs,  .  .  .  are  positive  constants  and  F,„  is  a  constant  ^  0.  A 
common  divisor  (involving  x)  of  Fj-i  and  F,-  would  divide  Fi-o,  .  .  .  , 
Fi,  Fi,  f,  /i,  whereas  f(x)  =  0  has  no  multiple  roots.  Hence  if  p  is  a  root 
of  Fi{x)  =  0,  then  F,-i(p)  t^  0  and 

Ci+iFi_i(p)  =  -/vi+i(p)  Fi+i{p),     c,>i  >  0,     ki+i{p)  >  0. 

Thus  Fi-i  and  Fi+i  have  opposite  signs  for  x  =  p.     We  proceed  as  in  §  4. 

Example  1.  If  f{.v)  =  .r^  +  6  x  —  10,  /i  =  3  (x-  +  2)  is  always  positive. 
Hence  we  may  employ  /  and  Fi  =  1.  For  x  =  —  oo,  there  is  one  variation  of 
signs;  forx=  +x,  no  variation.  Hence  there  is  a  single  real  root ;  it  lies  between 
1  and  2. 

Example  2.     U  fix)  =  2  x^  —  13  x-  —  10  x  —  19,  we  may  take 

/i  =  4  x^  -  13  X  -  5.. 
Then 

2f=xfi-h    h=  13x-+15x  +  38=  13(x+U)^+HI^ 

*  Usually  we  would  require  that  ki  be  positive  for  all  values  of  x,  since  we  usually 
wish  to  employ  the  limits  —  oo  and  -{-cc. 


§  6]  ISOLATION  OF  REAL  ROOTS  99 

Since  jz  is  always  positive,  we  need  go  no  further  (we  may  take  F2  =  1).     For 

X  =  —<x,  the  signs  are  -\ h;    for  x  =  +»,  H — h  +•      Hence  there  are  two 

real  roots.     The  signs  for  x  =  0  are  — \-.     Hence  one  real  root  is  positive  and 

the  other  negative. 

EXERCISES 

Isolate  by  Sturm's  theorem  the  real  roots  of 

1.   x^  +  3  x-  -  2  .c  -  5  -  0.  2.   X*  +  12  a;2  4-  5  a;  -  9  =  0. 

3.   .r^  -  7  x  -  7  =  0.  4.   3  x"  -  6  x"-  +  8  X  -  3  =  0. 

5.  x6  +  6  x^  -  30  x2  -  12  X  -  9  -  0  [stop  with  /2]. 

6.  x-i  -  8x^  +  25 x2  -  36 X  +  S  =  0. 

7.  For  /  =  x3  +  p.t  +  q  ip  ^  0),    /i  =  3  x^  +  ;>,    U  =  -2  p.c  -  3  g, 

4  fj,  ==  (-6  2;x  4-  9  ci)U  -  h    U  =  -4  p'  -  27  q% 

so  that  /s  is  the  discriminant  A  (Ch.  HI,  §  3).  Let  [p]  denote  the  sign  of  p.  Then 
the  signs  of/,  /i,  ft,  fs  are 

-      4-      +  [p]     [A]     for  X  =  -cc, 

+      +      -  [p]     [A]     for  X  =  4-^. 

For  A  negative  there  is  a  single  real  root.  For  A  positive  and  therefore  p  negative, 
there  are  3  distinct  real  roots.  For  A  =  0,  /2  is  a  divisor  of  /i  and  /,  so  that 
X  =  —  3  g/(2  /;)  is  a  double  root. 

8.  If  one  of  Sturm's  functions  has  p  imaginary  roots,  the  initial  equation  has  at 
least  p  imaginary  roots.     (Darboux.) 

6.   Sturm's  Functions  for  a  Quartic  Equation.     For  the  reduced  quar- 
.tic  equation /(2)  =  0, 

/  =  z'^  -\-  qz"^  -{-  rz  -\-  s, 

(5)  •   f,^4:z'^2qz  +  r, 

/o  =  —  2qz'  —  3rz  —  4:S. 

Let  q  9^  0  and  divide  cffi  by  /o.     The  negative  of  the  remainder  is 

(6)  U  =  Lz-  12 rs  -  rq^,     L  =  Sqs-2q^-  9r\ 

Let  L  5^  0.  Then  f^  is  a  constant  which  is  zero  if  and  only  if  /  =  0  has 
multiple  roots,  i.e.,  if  its  discriminant  A  is  zero.  We  therefore  desire  f^ 
expressed  as  a  multiple  of  A.     By  Ch.  IV,  §  4, 

(7)  A  =  -4  P3  _27  Q^     P  =  -4  s  -  |,     Q  =  §  gs  -  r^  -  ^^^  ^. 


100  THEORY  OF  EQUATIONS  [Ch.  IX. 

We  may  employ  P  and  Q  to  eliminate 

(8)  4s=-P-|',    r=== -Q-igP-Zzg'. 
We  divide  L'^f^  by 

(9)  f,  =  Lz  +  3rP,     L^9Q  +  4:qP. 
The  negative  of  the  remainder  is 

(10)  18  r2gP2  -  9  r'^LP  +  4  sL^  =  r/A. 

The  left  member  is  easily  reduced  to  q~\.     Inserting  the  values  (8)  and 
replacing  L^  by  L(9  Q  +  4  qP),  we  get 

- 18  qQP'-  -  12  g2p3  _  y  ^4  p2  _^  2  qP'~L  +  S  r/PL  -  3  q'-QL. 
Replacing  L  by  its  value  (9),  we  get  q-1.     Hence  we  may  take 

(11)  /.  =  A. 

Hence  if  qLl  7^  0,  we  may  take  (5),  (9),  (11)  as  Sturm's  functions. 
Denote  the  sign  of  q  by  [g].     The  signs  of  Sturm's  functions  are 

+     -      -  {q\     -  \L\     [A]     for  X  =  -  X, 
+     +      -[?]  {lA     [A]     for  .r  =  +  X. 

First,  let  A  >  0.  If  </  is  negative  and  L  is  positive,  there  are  four  real 
roots.  In  each  of  the  remaining  three  cases  for  q  and  L,  there  are  two 
variations  of  signs  in  either  of  the  two  series  and  hence  no  real  root. 

Next,  let  A  <  0.  In  each  of  the  three  cases  in  which  q  and  L  are  not 
both  positive,  there  are  three  variations  of  signs  in  the  first  series  and  one 
variation  in  the  second,  and  hence  just  two  real  roots.  If  q  and  L  are 
both  positive,  the  number  of  variations  is  1  in  the  first  series  and  3  in  the 
second,  so  that  this  case  is  excluded  by  the  Corollary  to  Sturm's  Theorem. 
To  give  a  direct  proof,  note  that  by  the  value  of  L  in  (6),  4  s  >  g^,  and 
that  P  is  negative  by  (7),  so  that  each  term  of  (10)  is  =  0,  whence  A  >  0. 

Hence,  if  gLA  5^  0,  there  are  four  distinct  real  roots  if  and  only  if  A 
and  L  are  positive,  and  q  negative;  two  distinct  real  and  two  imaginary 
roots  if  and  only  if  A  is  negative.     See  Ex.  5  below. 

EXERCISES 

1.  If  f/A  5^  0,  L  =  0,  then  /s  =  3  rP  is  not  zero  and  its  sign  is  immaterial  in 
determining  the  number  of  real  roots:  two  if  5  <  0,  none  if  g  >  0.  By  (10), 
5  has  the  same  sign  as  A. 


§0]  ISOLATION  OF  REAL  ROOTS  101 

2.  U  rA  9^  0,  q  =  0,  obtain  —fs  by  substituting  z  =--  — 4  s/(3  i)  in  /..  .Show 
that  we  may  take/s  =  rA  and  that  there  are  just  two  real  roots  if  A  <  0,  no  real 
root  if  A  >  0. 

3.  li  A  ^  0,  q  =  r  —  0,  there  are  just  two  real  roots  if  A  <  0,  no  real  root  if 
A  >  0.     Since  A  =  256  s^,  check  by  solving  z*  -\-  s  =  0. 

4.  If  A  5^  0,  qL  =  0,  there  are  just  two  real  roots  if  A  <  0,  no  real  root  if 
A  >  0.     [Combine  the  results  in  Exs.  1-3.] 

5.  If  A  <  0,  there  are  just  two  real  (distinct)  roots;  if  A  >  0,  g  <  0,  L  >  0, 
four  distinct  real  roots;  if  A  >  0  and  either  g  =  0  orL  =  0,  no  real  root.  [Com- 
bine the  theorem  in  the  text  with  that  in  Ex.  4.] 

6.  Apply  the  criterion  in  Ex.  5  to  Exs.  2,  4,  6,  p.  99. 

7.  Apply  to  Exs.  1-3,  p.  39,  and  Exs.  1-4,  p.  43. 

8.  Show  that  the  criterion  of  Ex.  5  is  equivalent  to  the  theorem  in  Ch.  IV,  §  7. 
If  A  >  0,  L  >  0,  g  <  0,  then  4  s  -  g^  <  0  by  (6).  Conversely,  if  A  >  0,  g  <  0, 
4  s  -  g2  <  0,  then  L  >  0.  For,  if  L  =  0,  9  Q  =  -4  gP  <  0,  since  P  <  0  by  the 
value  (7)  of  A.     Thus  81  Q^  S  16  q'~P'-,  A  =  8,  where 

5  =  -4  P3  _  i_(i  g2p2  =  4  P2(-  P  -  1  g2)  =  4  P2(4  s  -  g2)  <  0, 

— P  having  been  replaced  by  its  value  in  (7).  Thus  A  <  0,  contrary  to  hypothesis. 
The  two  criteria  for  four  real  roots  are  therefore  equivalent.  The  criterion  for 
2  distinct  real  and  2  imaginary  roots  is  A  <  0  in  each  theorem.  By  formal  logic 
the  criteria  for  no  real  root  must  be  equivalent. 

9.  If  a,  /3,  7  are  the  roots  of  a  cubic  equation  f(x)  —  0,  Sturm's  functions* 
/)  /i>  h,  /s  equal  products  of  positive  constants  by 

{x-a){x-l3){x-y),     ^{x-l3)(x-y),    ^^{a  -  ^X  -  y) ,    {a  -  py{a- yY{0  -  y^. 

Why  is  it  sufficient  to  prove  this  for  a  reduced  cubic  equation? 

Take  /  as  in  Ex.  7,  p.  99.  Proof  is  needed  only  for  the  third  function.  In  it 
the  coefficient  of  x  equals  2  2a-  —  2  2a/3  =  —  6  p,  while  the  constant  is 

—  Sa^Y  +  6  a^y  =  —3  g  —  6  g, 

by  Ex.  1,  p.  64.     Thus  the  third  function  equals  3/2. 

10.  Sturm's  functions  for  any  equation  with  the  n  roots  a,  fi,  .  .  .  ,  ir,  oo  equal 
products  of  positive  constants  by 

(X  -  a)    ...    (.C  -  co),      S(.C  -0)    .    .    .    (X-  co),    ^{a  -  /3)2(x  -  y)    .    .    .    {x  -  c.), 
2(a  -  py-ia  -  yYiff  -  yYi^  -  8)    .    .    .    {x  -  c.),    .    .    .    ,      («  -  ^)2   .    .    .    (^  _  „)2. 

Verify  this  for  n  =  4,  using  §  6.  A  convenient  reference  to  a  proof  for  any  n  is 
Salmon's  Modern  Higher  Algebra,  pp.  49-53. 

11.  There  are  as  many  pairs  p  of  imaginary  roots  as  there  are  variations  of 
signs  in  the  leading  coefficients  of  Sturm's  functions,  i.e.,  p  =  F+«,.  Hints: 
Of  any  two  consecutive  Sturm's  functions,  the  one  of  even  degree  has  the  same 
signs  for  X  =  —  00  and  x  =  +  00 ,  while  the  one  of  odd  degree  has  opposite  signs. 

*  In  Exs.  &-12,  it  is  assumed  that  there  are  ?i  +  1  Sturm's  functions  for  the  equatiou 
of  degree  n. 


102  THEORY  OF  EQUATIONS  [Ch.  IX 

Hence,  for  the  two  £uiiGtions,  F-c«  +  V+m  =  1.  There  are  n  pairs  of  coiisecu- 
tive  Sturm's  functions. . 

Hanoe  F-to  +  V.+x  —  n,  the  degree  of  the  equation. 

Subtract  V-co  —  I  +«=  =  '''>  ^'^^  number  of  real  roots. 

Thus  2  r+oo  =  n  -  r  =  2  p. 

12.  B}'  Exs.  10,  11,  the  number  of  pairs  of  imaginary  roots  is  the  number  of 
variations  of  signs  in  the  series 

1,  n,  2(a  -  /3)^      !(«  -  /3)-(a  -  7)^/?  "  7)',    •    •    •   , 

provided  no  one  of  these  sums  is  zero. 

7.t  Sturm's  Theorem  for  the  Case  of  Multiple  Roots.  Let*/„(.r)  be 
the  greatest  common  divisor  of  f{x)  and  /i  =  f'{x).  We  have  equations 
(1)  in  which  /„  is  now  not  a  constant.  The  difference  Va  —  Vi  is  the  num- 
ber of  real  roots  between  a  and  b,  each  multiple  root  being  counted  only  once. 

If  p  is  a  root  of  Ji{x)  =  0,  but  not  a  multiple  root  of  /(x)  =  0,  then 
fi-i{p)  7^  0.  For,  if  it  were  zero,  x  —  p  would  by  (1)  be  a  common  factor 
of/  and/i.     We  may  now  proceed  as  in  the  second  case  in  §  4. 

The  third  case  requires  a  modified  proof  only  when  r  is  a  multiple  root. 
Let  r  be  a  root  of  multiplicity  m,  m  ^  2.  Then/(r),  /'(r),  .  .  .  ,  /('«-')(r) 
are  zero  and,  by  Taylor's  Theorem, 

fir  +  p)  =  ^  .  ^  ^'".  .  ^^^  /^-KO  +  •  •  •  , 

These  have  like  signs  if  p  is  a  positive  number  so  small  that  the  signs  of 
the  polynomials  are  those  of  their  first  terms.  Similarly,  f{r  —  p)  and 
f'{r  —  p)  have  opposite  signs.  Hence  /  and  /i  show  one  more  variation  of 
signs  for  x  =  r  —  p  than  for  x  =  r  -\-  p.  Now  {x  —  r)"""^  is  a  factor  of 
/and/iandhence,  by  (1),  of/2,  .  .  .  ,  fn-  Let  their  quotients  by  this  factor 
be  (f>,  4>i,  .  .  .  ,  (i)n-  Then  equations  (1)  hold  after  the  /'s  are  replaced  by 
the  0's.  Taking  p  so  small  that  </>i(a:)  =  0  has  no  root  between  r  —  p  and 
r  -\-  p,  we  see  by  the  first  and  second  cases  in  §  4  that  </>i,  .  .  .  ,  </>„  show 
the  same  number  of  variations  of  signs  for  a;  =  r  —  p  as  for  a;  =  r  +  p. 
The  same  is  true  for  /i,  .  .  .  ,  /„  since  the  products  of  (^1,  ...  ,  ^n  by 
{x  —  r)""^  have  for  a  given  x  the  same  signs  as  </>!...,  <^„  or  the  same 
signs  as  —  (/>i,  .  .  .  ,  —</>„.  But  the  latter  series  evidently  shows  the 
same  number  of  variations  of  signs  as  ^i,  .  .  .  ,  <^„.  Hence  (4)  is  proved 
and  consequently  the  present  theorem. 

*  The  degree  of  /(x)  is  not  n,  nor  was  it  necessarily  n  in  §  4. 


§81  ISOLATION  OF  REAL  ROOTS  103 

EXERCISES 

l.f  For/  =  x'  -  8  .1-2  +  IG,  U  =  -i"'  -  ^  -i-,  h  =  -c-  -  4,  /i  =  a/2.     Hence  w  =  2. 
Then  F_oo  =  2,  F*  =  0,  and  there  are  only  two  real  roots,  each  a  double  root. 
2.t  /  =  (.c  -  \Y{x  -  2).         3.t    {x  -  \Y{x  +  2)^         4.t   x^  -  a;^  -  2  x  +  2. 

S.f  Budan's  Theorem.  Lef  a  and  b  be  real  numbers,  a  <  b,  neither 
a  root  of  f{x)  =  0,  ayi  equation  of  degree  n  with  real  coefficients.  Let  Va 
denote  the  7iumber  of  variations  of  signs  of 

(12)  fix),    fix),    f"{x),  .  .  .  ,    fi"){x) 

for  X  =  a,  after  vanishing  terms  have  been  deleted.  Then  Va  —  Vb  is  either 
the  nu7nber  of  real  roots  of  f{x)  =  0  between  a  and  b  or  exceeds  the  number  of 
those  roots  by  an  even  integer.  A  root  of  multiplicity  m  is  here  counted  as  m 
roots. 

In  case  Va  —  Vb  is  0  or  1,  it  is  the  exact  number  of  real  roots  between  a  and  b. 
In  other  cases,  it  is  merely  an  upper  limit  to  the  number  of  those  roots.  While 
therefore  the  present  method  is  not  certain  to  lead  to  the  isolation  of  the  real 
roots,  it  is  simpler  to  apply  than  Sturm's  method.  Indeed,  for  an  equation  of 
degree  6  or  7  with  simple  coefficients,  Sturm's  functions  may  introduce  numbers  of 
50  or  more  figures. 

The  proof  is  quite  simple  if  no  term  of  the  series  (12)  vanishes  for 
re  =  a  or  for  X  =  6  and  if  no  two  consecutive  terms  vanish  for  the  same 
vdue  of  x  between  a  and  b.  Indeed,  if  no  one  of  the  terms  vanishes  for 
Xi  ^  X  ^  X2,  then  Vx,  =  F^-,,  since  any  term  has  the  same  sign  for  x  =  Xi 
as  for  X  =  Xi.  Next,  let  r  be  a  root  of  f^'^(x)  —  0,  a  <  r  <  b.  By  hy- 
pothesis, the  first  derivative  f^'~^^^(x)  of  f-'^ix)  is  not  zero  for  x^r.  As  in 
the  third  step  (now  actually  the  case  i  =  0)  in  §  4,  /^'K^)  and  /'^'+^)(^)  show 
one  more  variation  of  signs  for  .r  =  r  —  7?  than  for  x  =  r  -\-  p,  where  p  is 
a  sufficiently  small  positive  number.  If  i  >  1,  /(')  is  preceded  by  a  term 
/(i-i)  in  (12).  By  hypothesis,  f'-'~'^'>(x)  5^  0  for  x  =  r  and  hence  has  the 
same  sign  for  x  =  r  —  p  and  x  =  r  -\-  p  when  p  is  sufficiently  small. 
For  these  values  of  x,  /(*K-^)  h^-s  opposite  signs.  Hence  f^'~^^  and  /(*') 
show  one  more  or  one  less  variation  of  signs  for  x  =  r  —  p  than  for 
X  =  r  +  p,  so  that  /^*~'^  f^'\  /^'+^^  show  two  more  variations  or  the 
same  number  of  variations  of  signs. 

Next,  let  no  term  of  the  series  (12)  vanish  for  a:  =  a  or  for  a;  =  6,  but 
let  several  successive  terms 

(13)  /('X^),   f^'+'Kx),  .  .  .  ,  f^'+^--'Kx) 


/(i) 

/(i+1) 

/(H 

h 

(-1)/ 

(-l)-l 

(-1) 

h 

+ 

+ 

+ 

104  THEORY  OF  EQUATIONS  [Ch.  ix 

all  vanish  for  a  valiio  r  of  x  between  a  and  h,  while /('+'^  (r)  is  not  zero, 
say  positive.*  Let  U  be  the  interval  between  r  —  p  and  r,  and  lo  the 
interval  between  r  and  r  -\-  p.  Let  the  positive  number  j)  be  so  small 
that  no  one  of  the  functions  (13)  or  /('+^)(x)  is  zero  in  these  intervals,  so 
that  the  last  function  remains  positive.  Hence  /('+'~')(a;)  increases  with 
X  (since  its  derivative  is  positive)  and  is  therefore  negative  in  /i  and  positive 
in  h.  Thus  Z^'+'^-K^)  decreases  in  /i  and  increases  in  1-2  and  hence  is 
positive  in  each  interval.  In  this  mamier  we  may  verify  the  signs  in  the 
following  table: 

.     .     .       /(''+J-3)       /('+/-2)       /(*•+/-!)    f(i+i) 
...        -  +  -  + 

.     .     .        +  +  +  + 

Hence  these  functions  show  j  variations  of  signs  in  7i  and  none  in  /o. 

If  i  >  0,  the  first  term  of  (13)  is  preceded  by  a  function  /('"^^(x)  which 
is  not  zero  for  x  =  r,  and  hence  not  zero  in  /i  or  I2  if  p  is  sufficiently  small. 

If  J  is  even,  the  signs  of /('~^)  and/(')  are  +  +  or 1-  in  both  /i  and  L2, 

showing  no  loss  in  the  number  of  variations  of  signs.  If  j  is  odd,  their 
signs  are 

/i      +  - 

or 

/2  +   +  -    + 

SO  that  there  is  a  gain  or  loss  of  a  single  variation  of  signs.     Hence 

show  a  loss  of  j  variations  of  signs  if  j  is  even,  and  a  loss  of  j  ±  1  if  j  is 
odd,  and  hence  always  a  loss  of  an  even  number  =  0  of  variations  of 
signs. 

If  i  =  0,  /(^)  =  /  has  r  as  a  j-fold  root  and  the  functions  in  the  table  show 
j  more  variations  of  signs  for  a:  =  r  —  p  than  ior  x  =  r  -}-  p. 

Thus,  when  no  one  of  the  functions  (12)  vanishes  for  x  =  a  or  for  .r  =  b, 
the  theorem  follows  as  at  the  end  of  §  4  (with  unity  replaced  by  the  mul- 
tiplicity of  a  root) . 

Finally,  let  one  of  the  functions  (12),  other  than /(.r)  itself,  vanish  for 
x  =  a  or  for  x  =  b.  If  5  is  a  sufficiently  small  positive  number,  all  of  the 
N  roots  of  f{x)  =  0  between  a  and  6  lie  between  a  +  5  and  b  —  5,  and 

*  If  negative,  all  signs  in  the  table  below  are  to  be  changed;  but  the  conclusion  holds. 


§9]  ISOLATION  OF  REAL  ROOTS  105 

for  the  latter  values  no  one  of  the  functions  (12)  is  zero.     By  the  above 
proof, 

Va+s  -  Vo-'  =  N-{-2t, 

where  t,  j,  s  are  integers  ^  0.     Hence  Va  —  Vb  =  N  -\-  2  (t  -\-  j  -\-  s). 

Example.     For/  =  x^  —  7  x  —  7, 

/'  =  3.^2-7,    /"  =  6.r,    f"'  =  (j. 

There  is  one  variation  of  signs  for  x  =  3,  but  none  for  x  =  4,  so  that  just  one  real 
root  lies  between  3  and  4.    For 

/  /'  /"  /'" 


-2 
-  1 


—  1  +5         —  12  +6  3  variations 

—  1  — 4         —    6  +6  1  variation. 


Thus  there  are  two  real  roots  or  no  real  root  between  —2  and  —1.  The  former  is 
the  case.  The  reader  should  isolate  the  two  roots  by  finding  an  intermediate  value 
of  X  for  which  the  series  shows  two  variations  of  signs. 

EXERCISES 

Isolate  by  Budan's  theorem  the  real  roots  of 

l.t  x?-x'^-2x  +  l=  0.  2.t  x^  +  3x'^-2x-5  =  0. 

3.t  If  /(a)  y^  0,  Va  equals  the  number  of  real  roots  >  a  or  exceeds  that  number 
by  an  even  integer. 

4.t  There  is  no  root  greater  than  a  number  making  each  of  the  functions  (12) 
positive,  if  the  leading  coefficient  of /(.r)  is  positive.     (Newton.) 

5.t   Divide /(x)  =  x"  +  aix"~^  +  •  •  •  by  x  —  a;   then 

fix)  =(x--a)l.c"-i  +  .i-"-V(«)+  •  •  •  +  gn-xicc)l+f{a), 

where  gi{a)  =«  +  «!,  giia)  =  «-  +  cha  +  a2,  .  .  .  .  If  a  is  chosen  so  that 
^i(a),  .  .  .  ,  Qn-iiot),  /(a)  are  all  positive,  no  positive  root  of  f{x)  =  0  exceeds  a. 
(Laguerre.) 

9.  Descartes'  Rule  of  Signs.  The  number  of  positive  roots  of  an 
equation  with  real  coefficients  either  equals  the  number  V  of  variations  of 
signs  in  the  series  of  coefficients  or  is  less  than  V  by  an  even  integer.  A  root 
of  multiplicitij  m  is  here  counted  as  m  roots. 

For  example,  x^  —  Sx'^-\-x-\-l  =  0  has  either  two  or  no  positive 
roots,  the  exact  number  not  being  found.  But  —  3  x^  -{-  x  +  I  =  0  has 
exactly  one  positive  root. 


106  THEORY  OF  EQUATIONS  ICh.  ix 

Consider  any  equation  with  real  coefficients 

f{x)  =  aox"  +  aix"-^  +   •  •  •  +  a„_i.r  +  a„  =  0, 
with  a„  5^  0.     For  x  =  0  the  functions  ( 1 2)  have  the  same  signs  as 

dn,    dn-l,     .     .     .     ,     fll,     Go, 

SO  that  Vo  =  V.  For  a:  =  +  x ,  the  functions  have  the  same  sign  (that 
of  ao) .  Thus  Vo  —  V^  —  V  is  either  the  number  of  positive  roots  or 
exceeds  that  number  by  an  even  integer.  Next,  the  theorem  holds  if 
/(O)  =  0,  as  shown  by  removing  the  factors  x. 

Corollary.  The  number  of  negative  roots  of  f{x)  =  0  is  either  the 
number  of  variations  of  signs  in  the  coefficients  of  f(  —  x)  or  is  less  than 
that  number  by  an  even  integer. 

Thus  x^  —  S  x^  -\-  X  -\-  1  =0  has  either  two  or  no  negative  roots,  since 
x^  —  Z  x^  —  x  -\-  1  =0  has  two  or  no  positive  roots. 

EXERCISES 

1.  x^  —  3x  +  2  =  0  has  one  negative  root  and  two  equal  positive  roots, 

2.  2^  -\-  a^x  +  6^  =  0  has  two  imaginary  roots  if  6  ?^  0. 

3.  For  n  even,  x"  —  1  =  0  has  only  two  real  roots. 

4.  For  n  odd,  x"  —  1  =  0  has  only  one  real  root. 

5.  For  n  even,  x"  +  1  =  0  has  no  real  root;  for  n  odd,  only  one. 

6.  x**  +  12  x^  +  5  X  —  9  =  0  has  just  two  imaginarj'^  roots. 

7.  x**  +  aV  +  6'X  —  c-  =  0  (c  9^  0)  has  just  two  imaginary  roots. 

8.  To  find  an  upper  limit  to  the  number  of  real  roots  of /(x)  =  0  l:)ctween  o  and  b,  set 

a  +  by 

X  = 


1+2/  \       ■"       b-x, 

multiply  by  (1  +  ?/)",  and  apply  Descartes'  Rule  to  the  resulting  equation  in  y. 

10.  t  Fourier's  Method.  If  Budan's  Theorem  gives  a  loss  of  two  or 
more  variations  of  signs  in  passing  from  a  to  a  larger  value  b,  and  hence 
leaves  in  doubt  the  number  of  real  roots  between  a  and  b,  we  may  employ 
a  supplementary  discussion. 

First,  let  /,  /',  /"  show  two  variations  of  signs  at  a  and  no  variation  at  b, 
while  the  series  beginning  wnth  /"  shows  no  loss  in  variations  (as  in  the 
Example  in  §  8).     Then  /"  is  of  constant  sign  between  a  and  b,  and  the 


§  101  ISOLATION  OF  REAL  ROOTS  107 

graph  oi  y  =  f{x)  has  a  (single)  maximum  or  minimum  point  between 
a  and  b,  according  as/"  is  negative  or  positive.     If  the  sum 

m  _  m 

of  the  subtangents  at  the  points  with  the  abscissas  a  and  6  is  >  h  —  a, 
the  tangents  cross  before  meeting  the  x-axis  and  the  graph  does  not  inter- 
sect the  X-axis  between  a  and  h,  so  that  there  are  two  imaginary  roots 
in  view  of  Budan's  Theorem  and 

(14)       71  =  v_^-v^  =  (7_.^  -  Va)  +  (Va  -  V,)  +:(n- VJ. 

In  the  contrary  case,  we  examine  the  value  half  way  between  a  and  h, 
etc.  Clearly  the  case  of  imaginary  roots  will  disclose  itself  after  a  very 
few  such  steps. 

Next,  in  the  general  case,  we  shall  find,  after  a  suitable  subdivision  of 
the  interval,  three  consecutive  functions 

/(/),     fU+i)^     /(;+2) 

showing  two  variations  of  signs  at  a'  and  no  variation  at  b',  while  the 
later  terms  of  the  series  show  no  loss  in  variations  of  signs.  We  may 
therefore  decide  as  in  the  first  case  whether  there  are  two  real  roots  of 
fU)  =  0  in  the  interval  [a',  b']  or  not,  and  in  the  latter  alternative  conclude 
that/  =  0  has  two  imaginary  roots.* 

Example.    Let          f{x)  =  x'  -  5x*  -  16 x^  +  12 x^-  -  9 .c  -  5.    Then 
fix)  =  5  x"  -  20  x3  -  48 .1-2  +  24  X  -  9, 
-i-/"(x)  =  5  x3  -  15  x2  -  24  X-  +  6, 
tV/'"(.c)  =5.t2-  lOx-8, 
jhf""{x)  -  X  -  1,    PK.C)  =  120. 

There  is  just  one  real  root  in  each  of  the  intervals  (—  3,  —2),  (  —  1,0),  (7,  8).  The 
interval  (0,  1)  is  in  doubt,  the  signs  being 

-  -     +     -     -     +     for  x-  =  0, 

—  —     —     —     —     +     for  X  =  1, 

where  0  is  read  — .    The ./  of  the  text  is  here  1 .     Now 

nil  _  I'M  =      -48      .    JL  =  3       3 
/"(I)      /"(O)       4  (-28)  ^4(6)       7'^8        ' 

*  For  further  details,  see  Serret,  Algcbre  Superieure,  ed.  4,  I,  pp.  305-31S. 


108  THEORY  OF  EQUATIONS  [Ch.  IX 

so  that  we  must  subdivide  the  interval.     For  x  =  ^,  the  signs  are  the  same  as 
for  x  =  1.    Thus  the  loss  in  variations  of  signs  occurs  in  the  interval  (0,  ^).     Now 

ni)   _  r(0)^  -  UtI-     1  3       1 
rU)       /"(O)       4(-9i)"^8      2 

Hence  there  are  two  imaginary  roots. 

EXERCISES 

l.t  x^  -  3  or*  +  2  x^  —  8  x2  +  3  .r  —  25  =  0  has  4  imaginary  roots. 
2.1   X®  +  x^  —  X*  —  x^  +  X-  —  X  +  1  =  0  has  G  imaginary  roots. 


CHAPTER  X 

Solution  of  Numerical  Equations 

1.   Newton's  Method.     To  find  the  root  between  2  and  3  of 

a:3  -  2  a:  -  5  =  0, 
Newton  *  replaced  re  by  2  +  p  and  obtained 

p3  +  6  p2  ^  10  p  -  1  =  0. 

Since  p  is  a  decimal,  he  neglected**  the  first  two  terms  and  set  10  p—  1  =  0, 
so  that  J)  =  0.1,  approximately.  Replacing  p  by  0.1  +  g  in  the  preceding 
cubic  equation,  he  obtained 

f/  +  6.3  g2  +  11.23  g  +  0.061  =  0. 

Dividing  —0.061  by  11.23,  he  obtained  —0.0054  as  the  approximate 
value  of  q.     Neglecting  (f  and  replacing  q  by  —0.0054  +  r,  he  obtained 

6.3  r2  +  11.16196  r  +  0.000541708  =  0. 
Dropping  6.3  r-,  he  found  r  and  hence 

rr  =  2  +  0.1  -  0.0054  -  0.00004853  =  2.09455147. 

This  value  is  in  fact  correct  to  the  seventh  decimal  place.  But  the 
method  will  not  often  lead  as  quickly  to  so  accurate  a  value  of  the  root. 

The  method  is  usually  presented  in  the  following  form.  Given  that  a 
is  an  approximate  value  of  a  real  root  of  fix)  =  0,  we  can  usually  find  a 
nearer  approximation  a  +  ^  to  the  root  by  neglecting  the  powers  h^,  h^,  .  .  . 
of  the  small  number  h  in  Taylor's  formula 

/(a  +  /0=/(a)+/'(a)/i+r(a)|+  .  .  . 
and  hence  by  taking 

f(a)+r(a)h  =  0,     h  =  ^j^' 

We  then  repeat  the  process  with  a  +  h  in  place  of  the  former  a. 

*  Isaaci  Newtoni,  Opuscula,  I,  1794,  p.  10,  p.  37  [found  before  1676], 
**  At  this  early  stage  of  the  work  it  is  usually  safer  to  retain  also  the  term  in  p^  and 
thus  find  p  approximately  by  solving  a  quadratic  equation. 

109 


110  THEORY  OF  EQUATIONS 

Thus  in  Newton's  example,  wc  have,  for  a  =  2, 


[Ch.  X 


h  = 


/'(2)    =I0'    «'  =  «  +  ^  =  2.1, 


-/(2.1)^  -0.061^  _ 
f'(2.1)  11.23 


0054 


2.  Graphical  Discussion  of  Newton's  Method.  Using  rectangular 
coordinates,  consider  the  graph  oi  y  =  fix)  and  the  point  P  on  it  with  the 
abscissa  OQ  =  a  (Fig.  22).     Let  the  tangent  at  P  meet  the  x-axis  at  T 


Q         T        T, 


Fig.  22 


Fig.  23 


and  let  the  graph  meet  the  x-axis  at  S.  Take  h  =  QT,  the  subtangent. 
Then 

QP=K<i),    f\a)  =  tan  XTP  =  -/(a)//i, 

f\a) 

In  the  fictitious  graph  in  Fig.  22,  OT  =  a  +  ^  is  a  better  approximation 
to  the  root  OS  than  OQ  =  a.  The  next  step  (indicated  by  dotted  lines) 
gives  a  still  better  approximation  OTi. 

If,  however,  we  had  begun  Avith  the  abscissa  a  of  a  point  Pi  near  a  bend 
point,  the  subtangent  would  be  very  large  and  the  method  would  probably 
fail  to  give  a  better  approximation.  Failure  is  certain  if  we  use  a  point 
P2  such  that  a  single  bend  point  lies  between  it  and  S. 

We  are  concerned  with  the  approximation  to  a  root  previously  isolated 
as  the  only  real  root  between  two  given  numbers  d  and  (8.  These  should 
be  chosen  so  nearly  equal  that/'(.r)  =  0  has  no  real  root  between  a  and  0, 
and  hence /(.r)  =  0  no  bend  point  between  a  and  /3.     Further,  if /"(x)  =  0 


SOLUTION  OF  NUMERICAL  EQUATIONS 


111 


has  a  root  between  our  limits,  our  graph  wiU  have  an  inflexion  point 
with  an  abscissa  between  a  and  jS,  and  the  method  hkely  will  fail  (Fig.  23) . 
Let,  therefore,  neither  J'{x)  nor  f"{x)  vanish  between  a  and  /3.  Since 
/"  preserves  its  sign  in  the  interval  from  a  to  /3,  while  /  changes  in  sign, 
/"  and  /  will  have  the  same  sign  for  one  end  point.  According  as  the 
abscissa  of  this  point  is  a  or  /3,  we  take  a  =  a  or  a  =  /3  for  the  first  step  of 
Newton's  process.  In  fact,  the  tangent  at  one  of  the  end  points  meets 
the  a;-axis  at  a  point  T  with  an  abscissa  within  the  interval  from  a  to  /3. 
If /'(x)  is  positive  in  the  interval,  we  have  Fig.  24  or  Fig.  25;  if/'  is  nega- 
tive, Fig.  26  or  Fig.  22. 


Fig.  24 


Fig.  25 


Fig.  26 


In  Newton's  example,  the  graph  l^etween  the  points  with  the  abscissas  a  =  2 
and  /3  =  3  is  of  the  type  in  Fig.  24,  but  more  nearly  like  a  vertical  straight  line. 
In  view  of  this  feature  of  the  graph,  we  may  safely  take  a  =  a,  as  did  Newton, 
although  our  general  procedure  would  be  to  take  a  =  0.  The  next  step,  however, 
accords  with  our  present  process;  we  have  a  =  2,  /3  =  2.1  in  Fig.  24  and  hence 
we  now  take  a  =  ^,  getting 

as  the  subtangent,  and  hence  2.1  —  0.0054  as  the  approximate  root. 

If  we  have  secured  (as  in  Fig.  24  or  Fig.  26)  a  better  upper  limit  to  the 
root  than  (8,  we  may  take  the  abscissa  c  of  the  intersection  of  the  chord 
AB  with  the  x-axis  as  a  better  lower  limit  than  a.     By  similar  triangles, 

-/(a)  :c-a=  m  :  /3  -  c, 

/n  _  am  -  I3f(a) 

^^  m-fioc)  ' 

This  method  of  finding  the  value  of  c  intermediate  to  a  and  /3  is  called  the 
method  of  interpolation  (regula  falsi). 


112  THEORY  OF  EQUATIONS  [Ch.  X 

In  Newton's  example,  a  =  2,  /3  =  2.1, 

/(«)  =  -1,    f{p)  =  0.061,     c  =  2.0942. 

The  advantage  of  having  c  at  each  step  is  that  we  know  a  close  limit  of 
the  error  made  in  the  approximation  to  the  root. 

We  may  combine  the  various  possible  cases  discussed  into  one: 

If  f(x)  =  0  has  a  single  real  root  and  f'{x)  =  0,  f"{x)  =  0  have  no  real  root 
between  a  and  /3,  and  if  we  designate  hy  (3  that  one  of  the  numbers  a  and  jS 
for  which  f (13)  and  f"  (13)  have  the  same  sign,  then  the  root  lies  in  the  narrower 
interval  from  c  to  (3  —  f{l3)/f'{(3),  where  c  is  given  by  (1). 

It  is  possible  to  prove*  this  theorem  algebraically  and  to  show  that  by 
repeated  applications  of  it  we  can  obtain  two  limits  a,  j3'  between  which 
the  root  lies,  such  that  a  —  ^'  is  numerically  less  than  any  assigned 
positive  number.  Hence  the  root  can  be  found  in  this  manner  to  any 
desired  accuracy. 

Example,    /(.c)  =  x^  -  2 .c^  -  2,     cc  =  2\,    /3  =  2^.    Then 

/(«)  =  -II,  m  =  h 

Neither  of  the  roots  0,  4/3  of /'(.r)  =  0  lies  between  a  and  /?,  so  that/(.r)  =  0  has 
a  single  real  root  between  these  limits  (Ch.  IX,  §  1).  Nor  is  the  root  §  of /"(.c)  =  0 
within  these  limits.  The  conditions  of  the  theorem  are  therefore  satisfied.  For 
a  <x  <  0,  the  graph  is  of  the  type  in  Fig.  24.     We  find  that 

c  =  Ml  =  2.349,     /3'  =  ^-^^  =2.3714, 

/3'-j^  =  2.3597. 

For  X  =  2.3593,  f{x)  =  -0.00003.  We  tliorc^fore  have  the  root  to  four  decimal 
places.     For  a  =  2.3593, 

/'(a)  =7.2620,     ^  -  jr^.  =2.3593041, 

which  is  the  valnc  of  the  root  correct  to  7  dociinal  i)laces.  For,  if  we  change  the 
final  digit  from  1  to  2,  tlie  result  is  greater  than  the  root  in  view  of  our  work,  while 
if  we  change  it  to  0,  f{x)  is  negative. 

*  Weber's  Algebra,  2d  ed.,  I,  pp.  380-382;  Kleines  Lehrbuch  der  Algebra,  1912,  p.  163. 


31 


SOLUTION  OF  NUMERICAL  EQUATIONS 


113 


EXERCISES 

(Preserve  the  numerical  work  for  later  use.) 

1 .  Find  the  root  between  1  and  2  of  x'  +  4  x^  —  7  =  0  correct  to  7  decimal 
places. 

2.  Find  the  root  between  —1  and  —2  to  5  decimal  places. 

3.  Find  a  root  of  x^  +  2  x  +  20  =  0  to  5  decimal  places. 

3.   Systematic  Computation  by  Newton's  Method.     Set 

/  1  /"       r  _  f"  _  1  r '       f  fill'  1  / ' 

J2—2J     }      J3  —  2     oJ        ~3y2,      •'^~2-S'4  ~  iJz  ,    •    '    '    * 

Then,  by  Taylor's  formula, 

fix  +  h)=  fix)  +  hfix)  +    h%(x)  +     h'Mx)  +     h%(x)  +  •  .  •  . 

fix  +  h)  =  fix)  +  2  hMx)  +  3  h%ix)  +  4  h%ix)  +  •  .  .  . 

f2{x  +  /O  =  f2(x)  +  3  /(/3(a:)  +  6  h'Mx)  +  •  •  .  . 

/3(a:  +  /i)  =  Mx)  +4/i/4(a;)  +  •  •  •  . 

The  second  formula  may  also  be  derived  from  the  first  by  differentiation 
with  respect  to  h  (or  if  we  prefer,  with  respect  to  x),  and  likewise  the 
third  from  the  second,  with  a  subsequent  division  by  2,  etc. 

Theworkof  finding /(x  + /;,),/' (x  +  /i),  .  .  .  horn  f{x) ,  f  (x) ,  f^ix) ,  .  .  . 
may  be  arranged  as  follows  for  the  case  7i  =  3,  whence /4  =  0: 


/3                       /2 

f 
+  /i(/2  +  hfs) 

/ 

+  h(r  +  /)/2  +  hj,) 

f  +  ¥2  +  hj, 

+  hih  +  2  /?/3) 

f+hf'-\-  h%  +  /i^a 
=  fix  +  /l) 

/2  +  2  hfs 

/'  +  2/2/2  +  3/1^3 

=  r(:r  +  A) 

/2  +  3/i/3      ^fziX  +  h) 

Here  we  have  added  /1/3  to  /o.  This  sum  is  multiplied  by  h  and  the 
product  added  to  /'.  To  the  resulting  sum  is  added  h  times  the  second 
sum  /2  +  2  hfs  in  the  second  column;  etc. 

Example  1.    f(x)  =  x^  -2x^  -  2.     Then 

fix)  =  3  x^  -  4  0-,    foix)  =  Sx-2,    f,ix)  =  1. 
Their  values  for  x  =  fi  =  2^  are  given  in  the  fii'st  line  below.     Since* 
h  =  -f/f  =  -0.129,  the  work  is  as  follows: 

*  Ordinarily  we  would  use  at  this  step  the  value  h  =—.13,  which  is  suflBciently 
exact  and  simplifies  the  numerical  work. 


114 


THEORY  OF  EQUATIONS 


[Ch.  X 


5.5 
-0.129 


8.75 
-0.69286 


1.125 
-1.03937 


5.371 
-0.129 


8.05714 
-0.67622 


0.08563 


5.242 
-0.129 


7.38092 


1  5.113 

The  numbers  at  the  bottom  arc  tlie  values  of 

h    M^'),    !'{&'),    /(^')     for /3' = /3 -f /i  =  2.371. 

Example  2.     Netto  treats  in  his  Algebra  the  equation 

f{x)  =  .r"  +  a;^  -  3  :i-2  -  a;  -  4  =  0. 
Then 

S'{x)  =  4  x^  +  3  .r^  -  6  a;  -  1,    /o  =  6  .r=  +  3  .r  -  3,    /s  =  4  .r  +  1,    /4  =  1. 

Since  /(I)  =  —6,  /(2)  =  6,  there  is  a  root  of  /(.r)  =  0  between  1  and  2.  By 
Descartes'  Rule, /'(.c)  and/2(.r)  each  have  a  single  positive  root.  Since /'(I)  =  0, 
/.(I)  =  6,  Ji{2)  =  27,  neither  has  a  root  between  1  and  2.  Since  /(2)  and  /"(2) 
are  of  like  sign,  we  take  p  =  2.  The  values  of  /4,  .  .  .  ,  /  for  .r  =  2  are  given 
in  the  first  hue  below. 

-«=-02 
31 


1 

9 

27 

31 

6 

-0.2 

-  1.76 

-  5.048 

-5.1904 

8.8 

25.24 

25.952 

0.8096 

-0.2 

-  1.72 

-  4.704 

8.6 

23.52 

21.248 

-0.2 

-  1.68 

8.4 

21.84 

-0.2 

8.2 
-0.04       -  0.3264 


-  0.860544       -0.81549824 


8.16 
-0.04 


21.5136 
-  0.3248 


20.387456 
-  0.847552 


8.12 
-0.04 


21.1888 
-  0.3232 


-0.8096 
21.248 
-0.04 


-0.00589824 


19.539904 


8.08 
-0.04 


20.8656 


8.04 


0.00589824 
19.539904 


=  0.000302- 


§4]  SOLUTION  OF  NUMERICAL  EQUATIONS  115 

The  root  is  2  -  0.2  -  0.04  +  0.000302  =  1.760302,  in  which  only  the  last  figure 
is  in  doubt.  Indeed,  it  can  be  proved  that  //  the  quotient  J/f  begins  with  k  zeros 
when  expressed  as  a  decimal,  the  best  approximation  is  obtained  by  carrying  the  division 
to  2  k  decimal  places. 

EXERCISES 

1.  Extend  the  work  of  Example  1  above. 

2.  Apply  the  present  method  to  Exs.  1,  2,  3,  page  113. 

3.  Treat  in  this  way  Newton's  example  (§  1). 

4.  In  the  four  long  formulas  at  the  beginning  of  §  3,  any  arithmetical  coefficient 
equals  the  sum  of  the  one  preceding  it  and  the  one  above  that  preceding  one,  as 
6-3  +  3,  4  =  1  +  3. 

4.   Horner's  Method.*     To  find  the  root  between  2  and  3  of 

a;3-2a;-5  =  0 

by  the  method  now  to  be  explained,  we  shall  modify  in  two  respects  the 
process  used  by  Newton  (§1).  While  in  the  latter  process  we  set  x  =  2  +  p 
and  found  the  cube  of  2  +  p,  etc.,  in  order  to  form  the  transformed  equation 

2)3  +  6  p2  +  10  p  -  1  =  0 

for  p,  we  shall  now  obtain  this  equation  by  a  different  process.  Since 
p  =  x-2, 

x'  -  2  X  -  5  ^  (x  -  2y  +  Q  (x  -  2y  +  10  {x  -  2)  -  1, 

identically  in  x.  Hence  —  1  is  the  remainder  obtained  when  x^  —  2  x  —  5 
is  divided  by  .r  —  2;   the  quotient  Q  evidently  equals 

(x  -  2)2  +  6  (x  -  2)  +  10. 

Similarly,  10  is  the  remainder  obtained  when  this  Q  is  divided  by  a:  —  2 
and  the  quotient  Qi  equals  (x  —  2)  -\-  6.  Another  division  gives  the 
remainder  6.  Hence  to  find  the  coefficients  6,  10,  —  1  of  the  terms  after 
p^  in  the  new  equation  in  the  variable  p  =  x  —  2,  we  have  only  to  divide 
the  given  function  x^  —  2  .r  —  5  by  a;  —  2,  the  quotient  Q  by  x  —  2,  etc., 
and  take  the  remainders  —1,  10,  6  in  reverse  order.  However,  when  the 
work  is  performed  as  tabulated  below,  no  reversal  of  order  is  needed, 
since  the  coefficients  then  appear  on  the  page  in  their  desired  order. 

*  W.  G.  Horner,  London  Philosophical  Transactions,  1819. 


116 


THEORY  OF  EQUATIONS 


[Ch.  X 


Synthetic  Division.     We  next  explain  a  brief  method  of  performing  a 
division  b}'  x  —  2  and,  in  general,  by  x  —  h.     AMien  we  divide 

fix)  =  aoa;"  +  aiX"'^  +•••+«„ 
by  X  —  h,  let  the  constant  remainder  be  r  and  the  quotient  be 

q{x)  =  box^-'  +  6iX"-2  +  .  .  .  +  bn-i. 

Comparing  the  coefficients  of  f{x)  with  those  in 
(x  -  h)  q(x)  +  r 

=  boX'^+ib,-hbo)x--'  +  (b2-hb,)x--''-\-  •  •  •  +(6„_i-/i6„_o)a:+r-;i6„_i, 
we  obtain  relations  which  may  be  written  in  the  form 

bo  =  ao,  6i  =  ai+/i6o,  b-y^ao+hbi,  .  .  . ,  6„_i  =  a„_i+/i6„_2,  ?-  =  a„+/i6„_i. 
The  steps  in  the  work  of  computing  the  6's  may  be  tabulated  as  follows: 


Oo 


hbo 


hbi 


a„_i 

hbn-2 


an 
hbn-1 


\h 


bo        61 


.     bn-1, 


In  the  second  space  below  Oo  we  write  60  (which  equals  Oo).  Then  mul- 
tiply 60  by  h  and  enter  the  product  under  ai,  add  and  write  the  sum  61 
below  it,  etc.  This  process  was  used  in  Ch.  I,  §  5,  to  get  the  value  r 
oifih).     See  also  Ch.  VI,  §6. 

In  our  example,  the  work  is  as  follows: 


1 

0 

-2 

-5 

2 

4 

4 

1 

2 

2 

-  1 

2 

8 

1 

4 
2 

10 

1         6 

Thus  1,  6,  10,  —1  are  the  coefficients  of  the  equation  in  p. 

But  there  is  a  more  essential  difference  between  the  methods  of  Horner 
and  Newton  than  the  detail  as  to  the  actual  work  of  finding  the  trans- 
formed equations.  Newton  used  the  close  approximation  0.1  to  the  root 
of  the  equation  in  p.     As  this  value  exceeds  the  root  p  and  hence  would 


J  4] 


SOLUTION  OF  NUMERICAL  EQUATIONS 


117 


lead  to  a  negative  correction  at  the  next  step,  Horner  would  have  used 
the  approximation  0.09  (taking  a  decimal,  with  a  single  significant  figure, 
just  less  than  the  root).  The  next  steps  of  Horner's  process  are  as 
follows : 

16  10  -1  10.09 


0.09 

0.5481 

0.949329 

6.09     10.5481 
0.09     0.5562 

-0.050671 
0.044517584 

6.18 
0.09 

11.1043 

0.05 
11.1 

6.27 

=  0.004 

0.004    0.025096 

6.274    11.129396 
0.004     0.025112 

-0.006153416 

6.278 
0.004 

11.154508 

1 


6.282 


Hence  a;  =  2.094+f,  where  t  is  between  0.0005  and  0.0006.  Thus  ^^+6.282  f 
is  between  0.0000015  and  0.0000023,  so  that  the  constant  term  should  be 
reduced  by  2  in  the  sixth  decimal  place.     We  now  have 

11.154508 «  =  0.006151  +  ,     t  =  0.0005514+, 
with  doubt  only  as  to  whether  the  last  figure  of  t  should  be  4  or  5. 

Example  1.     Find  the  root  between  1  and  2,  correct  to  seven  decimal  places,  of 
a;3  +  4  x2  -  7  =  0. 

See  p.  118.     The  figure  in  the  fourth  decimal  place  is  evidently  2.     Thus 
x  =  1.164  +  y,     0.0002  <y<  0.0003,     y'  +  7.492  ?/-+•••   =0, 

0.000000299  <  y^  +  7.492  //^  <  0.000000675, 
0.003316381  <  13.376688?/  <  0.003316757, 
0.00024792   <  y  <  0.00024795. 

Hence  x  =  1.1642479+,  in  which  all  of  the  figures  are  correct.  But  this  work  may 
be  abridged.  The  sum  of  the  terms  in  y^  and  y^  has  its  first  significant  figure  in  the 
seventh  decimal  place,  as  shown  by  7.5  (0.0003)-.  Hence,  returning  to  the  final 
numbers  in  our  transformation  scheme  above,  we  carry  the  division  of  0.0033170 
by  13.376688  until  we  reach  a  remainder  whose  sign  is  in  doubt  in  view  of  the 
doubt  on  the  seventh  decimal  place  of  the  dividend.     Doubt  would  hero  first  arise 


118 


THEORY  OF  EQUATIONS 


(Ch.  X 


-7 
5 


1 

5 

5 

-2 

1 

6 

1 

6 

1 

11 

7 
0.1 


10. 1 


0.71 


1.171 


1 

7.1 

11.71 

-0.829 

0.1 

0.72 

1 

7.2 
0.1 

12.43 

7.3 
0.06 


10.06 


0.4416 


0.772296 


7.36 
0.06 


12.8716 
0.4452 


7.42 
0.06 


-0.050704 


13.3168 


1         7.48 
0.004 


10.004 


0.029936         0.053386944 


1    7 .484 
0.004 


13.346736 
0.029952 


1         7.488 
0.004 


-0.003317056 


13.376688 


7.492 


in  the  case  of  the  figure  9  in  the  seventh  decimal  place  of  the  quotient;  but  this 
doubt  is  removed  by  noting  tliat  the  correction  to  lie  subtracted  from  tlie  seventh 
decimal  place  of  the  dividend  is  a  figure  lx't\v(H>n  2  and  7  (as  sliown  by  the  above 
examination  of  the  terms  in  //•''  and  y''). 

ExAMPLK  2.     Find   the  root  between    —4  and    —3,   (correct  to  seven  decimal 
places,  of  the  equation  in  E.x.  1. 

Using  the  multipliers  -4,  +0.6,   +0.008,  we  find  that  .r  =  -4  +  0.608  +  y 
where 

?/  -  6.176  if  +  7.380992?/  -  0.004556288  =  0. 

Thus  y  just  exceeds  0.0006.  The  sum  of  the  terms  in  y^  and  y-  is  -0.000002  to 
six  decimal  places.  Carr^nng  tlie  division  of  0.004558  by  7.3S1  until  the  sign  of 
the  remainder  is  in  doubt,  on  account  of  the  doubt  in  the  sixth  decimal  place,  we 


5  5,  6] 


SOLUTION  OF  NUMERICAL  EQUATIONS 


119 


get  y  —  0.0006175,  with  the  shght  doubt  due  to  the  approximate  value  of  the 
divisor  and  that  of  the  y-  term.  Since  the  cube  of  6.176  is  just  less  than  235.6 
(as  shown  by  logarithms),  the  sum  of  the  terms  in  y^  and  y'^  is  —0.000002356 
to  nine  decimal  places.  Cariying  out  the  division  of  0.004558644  by  the  exact 
coefficient  of  y,  we  get  y  =  0.0006176,  correct  to  seven  decimal  places.  Hence 
X  =  -3.3913823. 

EXERCISES 

1.  Find  to  7  decimals  the  root  of  .i-^  +  4  .r-  —  7  =  0   between  —1,  —2. 

2.  Find  to  7  decimals  all  the  roots  of  .c^  —  7  x  —  7  =  0. 
Find  to  5  decimals  all  tlie  real  roots  of 

3.  x^  +  2  a;  +  20  =  0.  4.  a;^  +  3  x^  -  2  a;  -  5  =  0. 

5.   x^  +  x'-2x-  I  =  0.  6.   x^  +  4:  x^  -  17.5  x^  -  18  a;  +  5S.5  =  0. 

7.  x*  -  11727  x  +  40385  =  0.     (G.  H.  Darwin.) 

8.  Find  to  8  decimals  the  root  between  2  and  3  of  x^  —  x  —  9  =  0  by  making 
only  three  transformations. 

5.t  Without  the  intermediation  of  the  idea  of  division  by  x  —  h,  we 
may  show  directly  that  the  process  of  §  4  yields  the  correct  transformed 
equation.     For  simplicity,  we  take  a  cubic  equation 

f(x)  =  ax^  -{-  hx^  -\-  ex  -\-  d  =  0. 

Our  process  was  as  follows: 

a         b  c  d  1^ 


ah 


ah"  +  hh 


ah^  +  fe/i2+  ch 


ah  +  h 
ah 


ah"^  +  6/i  +  c 
2  ah"  +  bh 


ah^  +  bh"  -\-ch-\-d 
=  f{h) 


2ah-{-b 
ah 


3  a/i2  +  2  6/i  +  c  =  f'(h) 


a      3ah  +  b     =  hf'Qi) 
Hence  the  transformed  equation  is 

\r{h)v'  +  \f"(ji)f  ^nh)p +/(/o  =  0. 

The  terms  of  the  left  member,  read  in  reverse  order,  are  those  of  Taylor's 
formula  for  the  expansion  of  f{h  -\-  p).  Hence  the  above  process  yields 
the  equation  obtained  from  f(x)  =  0  by  setting  x  =  h  -{-  p. 

6.t   Numerical  Cubic  Equations.     After  finding  a  real  root  r  5^  0  of 
fix)  =  x^ -\- bx"" -\- ex  ^  d  =  0, 


120  THEORY  OF  EQUATIONS  [Ch.  X 

we  may  o})tain  the  remaining  roots  ri  and  r^  from 

^1  +  »'2  =  —h  —  r,     Tiro  = =  r'  -j- br  -\-  c. 

We  have 

(2)  (ri  -  ro)^  =  (ri  +  ro)^  -  4  rira  =  &2  _  4  c  -  2  6r  -  3  r^. 

Thus  ri  —  ro  is  either  real  or  a  pure  imaginary.  Making  use  also  of 
^1  +  fo,  we  shall  have  the  real  or  imaginary  expressions  of  ri,  Vo.  As  it 
would  be  laborious  to  compute  the  right  member  of  (2),  we  may  make 
use  of  a  device.     We  have 

(ri-ro)2  =  62_3c_/'(r). 

The  value  of  f'{r)  for  the  approximate  value  of  /•  obtained  at  any  stage 
of  Horner's  process  is  the  coefficient  preceding  the  last  one  in  the  next 
transformed  equation  (§5). 

Example.     Let/(.r)  =  x^  +  4:X-  —  7.     By  Ex.  1,  p.  117, 

/'(1. 164)  =  13.376688. 

If  we  continue  Horner's  process,  using  the  multiplier  m  =  0.000248,  and  retaining 
only  six  decimal  places,  we  see  that  we  must  twice  add  7.492  m  =  0.001858  to  the 
preceding  /'  to  get 

/'(?•)  =  13.380404,     r  =  1.164248. 

But  this  continuation  of  Horner's  process  is  unnecessary.     Using  J"'{x)  =  6  and 

the  work  on  p.  118,  we  have 

fix  +  m)  =  fix)  +  mf'ix)  +  3  m^,     ^/"(1.164)  =  7.492, 
/'(r)  =  13.37  .  .  .  +  2  m  (7.492)  +  0.0000002  -  13.3804042. 

Hence  we  get 

(r,  -  r,)2  =  2.6195958,        n  -  ro  =  1.6185165, 
n  +  r,  =  -5.1642479,     n  =  -1.7728657,     r.  =  -3.3913822. 

Tf.   Numerical  Quartic  Equations.     Let 

fix)  =  X'  +  bx^  +  cx2  +  f/x  +  e  =  0 

have  two  distinct  real  roots  r  and  s.  When  these  arc  found  approximately 
by  Horner's  process,  we  get  at  the  same  time  f'{r),  f'is),  approximately. 
Call  the  remaining  roots  ri  and  ro.     Then 

>'i  +  ^2  =  —  6  —  r  —  s, 
TiTi  =  c  —  ir  -^  s)ii\  +  fo)  —  rs  =  c  -{-  bir  -\-  s)  -\-  r-  +  rs  +  s^, 
in  -  r.,)-  =  6''  -  4  c  -  2  b(r  +  s)  -  3  r^  -  2  rs  -  3  .s^, 
in  -  r,)'~ib  +  2  r  +  2  s)  =  -  in  -  r,y'i2  n  +  2  r,  +  b) 
=  6^  -  4  6c  -  8  c(r  +  &)+  6(  -  7  r-  - 10  rs  -  7  s-)  -  G  r^  - 10  r's  - 10  rs~  -  6  s\ 


§81  SOLUTION  OF  NUMERICAL  EQUATIONS  121 

To  the  second  member  add  the  jDroduct  of  10  by 

H  +  r^s  +  rs^  +  s^  +  6(r2  +  rs  +  s-)  -\- c(r -^  s)  +  d  =  -^MjIliM  =  q. 

r  --  s 

Hence 

(ri  -  r.Yib  +  2r  +  2s)=63-4  6c  +  8 d+fir)  +  /'(«)• 

From  this  equation  we  get  n  —  r^  and  then  find  Vi  and  ro,  approximately. 

EXERCISES  t 

1.  After  finding  one  of  the  real  roots  of  the  cubic  equations  in  Exs.  2,  3,  4,  5,  8, 
p.  119,  find  the  remaining  roots  by  §  6. 

2.  Treat  the  quartic  equations  in  Exs.  6,  7,  p.  119,  by  §  7. 
Find  two  and  then  all  of  the  roots  of 

3.  x^  +  12  X-  +  7  =  0.  4.  x'  -  80  x3  +  1998  x^  -  14937  x  +  5000  =  0. 

S.f  Graffe's  Method.  First,  let  all  of  the  n  roots  a:i,  .  .  .  ,  .r„  be  real 
and  distinct  numerically.  Choose  the  notation  so  that  Xi  exceeds  X2  nu- 
merically and  X2  exceeds  X3  numerically,  etc.     In 

(3)  2.rr  =  ..-(l+|,  +  |=+-- 

each  fraction  approaches  zero  as  m  increases,  so  that  Xi"  is  an  approxi- 
mate value  of  Sxi"*  if  ?w  is  sufficiently  large.     Similarly, 

(4)  2xi"'X2-  =  x^'^xr  (  1  +  ^  +  —  +•••  +  -^^^  + 


Xi'"-        Xi'"  Xi"'X2" 

so  that  Xi^Xi"^  is  an  approximate  value  of  liXi^x-f^  for  m  large.  Now 
x{^,  .  .  .  ,  Xn""  are  the  roots  of 

(5)  ?/"  -  So'i'"  •  y"-i  +  Sa;i'"X2'"  •  2/""^-   •  •  •  =  0. 

As  illustrated  in  the  examples  below,  it  is  quite  easy  to  form  this  equation 
(5)  for  values  of  m  which  are  the  successive  powers  of  2.  After  obtaining 
the  equation  in  which  m  is  sufficiently  large,  we  divide  each  coefficient 
by  the  preceding  coefficient  and  obtain  approximate  values  of  the  nega- 
tives of  Xi'",  X'f,  ....     Indeed,  the  coefficients  are  approximately 

1,  —  X{^,    X{^X-f,   —Xi'^X-^^^Xii^,    .... 

Example  1.     For  x^  -\r  x"^  —  2  x  —  \  —  Q,  we  first  form  the  cubic    equation 
whose  roots  are  the  squares  of  the  roots  x\,  x^,  xz  of  the  given  equation.    To  this 


122  THEORY  OF  EQUATIONS  [Ch.  X 

end,  we  transpose  the  terms  x"^,  —  1,  of  even  degree,  square,  replace  x-  by  y,  and  get* 

if  -  5y'-  +  Gij  -  1  =  0, 

whose  roots  are  v/i  =  xr,  ?/j  =  .r2-,  //a  =  x-^-.     Repeating  the  operation,  we  get 

^-'^  -  13  z-  +  26  2  -  1  =  0,     y^  -  117  ?;-  +  050  y  -  1  =  0, 

with  the  roots  Zi  =  y{',  .  .  .  ,  and  Vi  =  zr,  ....  Hence  the  roots  of  the  y-cubic 
are  the  8th  powers  of  .ri,  X2,  X3.  By  logarithms,  the  Sth  roots  of  117,  ff?,  sh  (the 
approximate  values  of  Xi^,  x-i^,  Xs^)  are  1.81.3,  1.239,  0.4-150,  which  are  therefore 
approximate  numerical  values  of  .ri,  x-z,  x-i.     The  next  step  gives  the  equation 

w'  -  12389  ic-  +  4222GG  10  -1=0. 

The  16th  roots  of  12389,  etc.,  are  - 1.80225,  1.24676,  -0.44504,  to  which  the  proper 
signs  have  now  been  prefixed  (their  product  being  positive  and  sum  being  —1). 

Instead  of  repeating  the  process,  we  may  now  obtain  as  follows  the  values  of  the 
roots  correct  to  five  decimal  places.  We  had  the  logarithms  of  the  last  approxi- 
mations to  the  roots  and  hence  see  at  once  that  {xs/x-iY^  affects  only  the  Sth  decimal 
place  and  that  (.r3/.ci)^^is  still  smaller.  The  coefficient  of  10  is  2xi^®.r2>^  whose 
expression  (4)  involves  only  the  first  three  terms.    Hence 

.nisrois  =  422266, 

correct  to  7  decimal  places.  The  reciprocal  is  .r3'^  whence  Xs  =  —0.44504  to 
5  decimal  places.  By  the  approximate  values  of  .ri  and  .r-.  from  the  it'-cubic, 
(xi/xiY^  =  0.002751.    Thus 

1.002751  .ri's  =  12389  =  S-ri", 
whence  Xi  =  —1.80194  to  5  decimal  places.     By  the  displayed  equations, 
.,      422266  X  1.002751 


12389 


.T2  =  1.24698. 


We  have  now  found  each  root  correct  to  five  decimal  places.     As  a  check,  note 
that  the  roots  are  (Ch.  VIII,  §  3,  §  8) 

27r        ^  Air        ^  67r 

2  COS—-,     2  COS-—,     2cos-;r- 

7  7  7 

The  above  process  requires  modification  if  several  of  the  largest  roots 
are  equal  or  approximately  equal  numerically.  If  Xi  and  Xo  are  approxi- 
mately equal,  but  sufficiently  different  from  x^,  .  .  .  ,  x^,  numerically,  an 
approximate  value  of  .ri™  is  2  ^-Yi'"- 

Next,  consider  a  cubic  equation  with  two  conjugate  imaginary  roots 

*  We  may  use  symmetric  functions:   2(/i  =  2xi-  =  (Sxi)-  —  2  S.C1X2  =  5,  etc. 


§  8]  SOLUTION  OF  NUMERICAL  EQUATIONS  123 

X2  and  xz,  whose  modulus  (Ch.  II,  §  8)  is  r,  and  a  real  root  Xi  numeri- 
cally greater  than  r.     Then  the  real  number 

xf^.x^ 

is  numerically  less  than  or  equal  to  the  sum 

^  /mod.  X2 

[     ±Xi 

of  the  moduli  of  its  two  parts,  and  hence  approaches  zero  as  m  increases. 
Thus,  by  (3),  an  approximate  value  of  Xi'"  is  2xi"'. 

Example  2.  For  .c^  -  2  a;  -  2  =  0,  Xi  >  1.7,  X2X3  =  r^  =  2/a;i.  Since  2  <  (1.7)^ 
r  <  1.7  <  Xi.  Forming  the  equation  whose  roots  are  the  squares  of  the  roots  of 
the  x'-cubic,  that  whose  roots  are  the  fourth  powers,  etc.,  we  get 

?/3  -  4  ?/2  +  4  7/  -  4  =  0, 
z'-Sz''-  IGz-  16  =  0, 
v^  -  96  i;2  -  256          =  0. 


Thus  .ri  is  approximately 


\/96  =  1.7692  . 


By  two  more  steps,  we  get 

a-i  =  ^^85032960  =  1.769293, 
correct  to  six  decimal  places. 

For  a  cubic  equation  in  which  Xi  <  r,  we  employ  the  equation  in  X 
obtained  by  setting  x  =  l/X.  Its  root  1/xi  exceeds  numerically  the  mod- 
ulus 1/r  of  the  imaginary  roots  l/Xi,  l/xs.  Hence  the  equation  in  X  is  of 
the  type  last  discussed. 

EXERCISES  t 

1.  The  equation  whose  roots  are  the  Sth  powers  of  the  roots  Xi,  X2,  X3  of 
x^  —  4  x''^  —  .r  +  3  =  0  is 

w^  -  74474  w^  +  46213  iv  -  6561  =  0. 

Dividing  the  negative  of  each  coefficient  by  the  preceding  coefficient  and  extracting 
the  Sth  root  of  each  quotient,  we  get  4.06443,  0.94,  0.78.  The  first  is  a  good 
approximation  to  Xi.  The  last  two  are  approximately  equal  and  hence  not  good 
approximations  to  —X2,  X3.  To  avoid  this  inconvenience,  add  unity  to  each  root 
(i.e.,  replace  x  by  X  —  1).  Treat  the  equation  in  X  and  so  obtain  good  approxi- 
mations to  Xi,  Xi,  X3. 


124  THEORY  OF  EQUATIONS  [Ch.  x 

Treat  by  the  present  methods 

2.   r'  -  2  a;  -  5  =  0.  3.   x''  -  2  x-  -  2  =  0.  4:.   t^  +  ix'-  -  7  =  0. 

5.   a;3  +  2  X  +  20  =  0. 

For  further  details  on  the  determination  of  imaginary  roots  by  this  method, 
see  Encke,  Crelle's  Journal,  vol.  22  (1841),  p.  193;  and  examples  by  G.  Bauer, 
Vorlesungen  ilber  Algebra,  1903,  p.  244;  and  C.  Runge,  Praxis  der  Gleichungen, 
1900,  p.  157. 

9.t  To  determine  the  imaginary  roots  of  an  equation  f{z)  =  0  with 
real  coefficients,  expand /(a:  +  yi)  by  Taylor's  Theorem;  we  get 

Kx)  +f'{x)yi  -  fix)   t^-  r{x)  ,^^^+  •  .  .   =0. 


Since  x  and  y  are  to  be  real,  and  y  9^  0, 
(6) 


m  -  fix)  ^ + r'ix)  ^ .  2^  3 . 4  -  •  •  •  =0, 


f(x)-nx)j^+fi^Kx)f,-  •  •  •  =0. 

By  eliminating  y^  between  these  two  equations,  we  obtain  an  equation 
E{x)  =  0,  whose  real  roots  x  may  be  found  by  one  of  the  preceding 
methods.  In  general  the  next  to  the  final  step  of  the  elimination  gives 
y-  as  a  rational  function  of  x,  so  that  each  real  x  which  yields  a  positive 
real  value  of  y^  furnishes  a  pair  of  imaginary  roots  x  ±  yi  of  f{z)  =  0.  But 
if  there  are  several  pairs  of  imaginary  roots  with  the  same  real  part  x,  the 
equation  in  y^  used  in  the  final  step  of  the  elimination  will  be  of  degree 
greater  than  unity  in  y-. 

Example.    For  f{z)  =  s-*  —  3  +  1,  equations  (G)  are 

x<  -  X  +  1  -  6  xhf  +  y^  -  0,     4  x^  -  1  -  4  xy^  =  0. 
Thus 

^'  "^  ^'  ~  4^ '    "  ^'''  "^  ''■'  "^  IB  "  ^• 

The  cubic  equation  in  x^  has  the  single  real  root 

x2  =  0.528727,    x  =  ±0.72714. 
Then  y^  =  0.87254  or  0.184912,  and 

z  =  x  +  yi  =  0.72714  ±  0.43001  i,     -  0.72714  ±  0.93409  i. 


10] 


SOLUTION  OF  NUMERICAL  EQUATIONS 
EXERCISES  t 


125 


1.  For  the  quartic  equation  in  Ch.  V,  §  1,  eliminate  i/  between  equations  X  =  0, 
Y  —  0,  corresponding  to  the  present  pair  (6),  and  get 

x{x  -  2)  (16  x'  -  64  x^  +  136  x-  -lUx  +  65)  =  0. 

Show  that  the  last  factor  has  no  real  root  by  setting  2  x  =  w  -{-  2  and  obtaining 
(«'2  +  l)(iv^  +  9)  =  0.  Hence  find  the  four  sets  of  real  values  x,  y  and  hence  the 
four  complex  roots  x  +  yi. 

2.  If  r  and  s  are  any  two  roots  of  f{z)  =  0  and  we  set 


r  -\-  s 


y 


2i 


we  have  r  =  x  -{-  yi,  s  =  x  —  yi,  so  that  f{x  ±  yi)  =  0.  Hence  E{x)  =  0  has 
as  its  roots  the  ^  n(n  —  1)  half-sums  of  the  roots  oif{z)  =  0  in  pairs.  If,  however, 
we  eliminate  x  between  equations  (6)  and  set  —  4  y^  =  ty,  we  obtain  an  equation 
in  w  whose  roots  are  the  §  /t(n  —  1)  squares  of  the  differences  of  the  roots  of/(2)  =  0. 

10 1-   Lagrange's  Method.     The  root  between  1  and  2  of 

x^  +  4  a;2  -  7  =  0 
may  be  expressed  as  a  continied  fraction.     Set  x  =  1  +  1/y.     Then  * 

-2i/-\-nif  +  7y-{-l  =  0. 
Since  —2i/-\-  11  y-  must  be  negative,  we  have  y  >  5.     We  find  by  trial 
that  y  lies  between  6  and  7.     Set  y  =  6  +  1/z. 


-2 

11 
-12 

7 
-   6 

1   [6 
6 

-2 

-    1 
-12 

1 

-78 

7 

-2 

-13 
-12 

-77 

•25 


2_25_77 


z"       z"-         Z 
Since  7  2^  -  77  ^^  >  0,  2  >  11. 
Now 

a;  =  1-f 


7  2'  -  77  22  -  25  2  -  2  =  0. 
The  value  of  z  lies  between  11  and  12. 

1  72  +  1 


6  + 


1      62  +  1 


*  We  may  of  course  first  set  x  =  1  +  rf,  find  the  cubic  equation  in  d,  by  our  earlier 
method,  and  then  replace  d  by  \ly. 


126  THEORY  OF  EQUATIONS  ICh.  X 

Using  z  =  11,  we  find  that  x  is  just  smaller  than  1.1642.     But  z  is  in  fact 
just  greater  than  11.3.     Using  z  =  11.3,  we  find  that 

Hence  a:  =  1.1642  to  four  decimal  places. 

There  is  a  rapid  method  of  evaluating  a  continued  fraction  and  a  means 
of  finding  the  limits  of  the  error  made  in  stopping  the  development  at  a 
given  place.  For  an  extensive  account  of  the  theory  and  applications  of 
continued  fractions,  see  Serret's  Coum  d'Algcbre  Superieure,  ed.  4,  I, 
pp.  7-85,  351-368. 


CHAPTER  XI 


(1) 


Determinants;  Systems  of  Linear  Equations 

1.   In  case  there  is  a  pair  of  numbers  x  and  y  for  which 

\aix  +  hiij  =  ki, 
\ch.x^-  hiij  =  ki, 

they  may  be  found  as  follows.  Multiply  the  members  of  the  first  equa- 
tion by  62  and  those  of  the  second  equation  by  —61,  and  add  the  resulting 
equations.     We  get 

(0162  —  a.<>h\)x  =  k^o  —  kihi. 

Employing  the  respective  multipliers  —ao  and  ai,  we  get 

(ttihi  —  a2hi)y  =  01^2  —  CL2ki. 
The  common  multiplier  of  x  and  y  is 

(2)  0162  -  aihi, 


which  is  called  a 

determinant 

of  the  second  order  and  denoted  by  the 

symbol  * 

(2') 

1  ai  61 
1  ao  62 

The  value  of  the  symbol  is  obtained  by  cross-multiplication  and  substrac- 

tion.     Our  earlier  results  now  give 

(3) 

a2  62 

X  = 

ki  61 
^-2  62 

1 

ai  hi 
a2  62 

y  = 

ai  ki 

02   /C2 

• 

We  shall  call  ki  and  ki  the  known  terms  of  our  equations  (1).  Hence, 
if  D  is  the  determinant  of  the  coefficients  of  the  unknowns,  the  'product  of  D  by 
any  one  of  the  unknowns  equals  the  determinant  obtained  from  D  by  substi- 
tuting the  known  terms  in  -place  of  the  coefficients  of  that  unknown. 

*  The  symbol  for  an  expression  should  show  explicitly  all  of  the  quantities  upon 
whose  values  the  value  of  the  expression  depends.  Here  these  are  ai,  61,  a<>,  h%.  The 
advantage  of  writing  these  in  the  symbol  (2')  in  the  order  in  which  they  occur  in  the 
equations  is  that  the  symbol  may  be  written  down  without  an  effort  of  memory  by  a 
mere  inspection  of  the  given  equations. 

127 


128 


THEORY  OF  EQUATIONS 


[Ch.  XI 


Example.    For  2  a;  —  3  y  =  —4,        6  x  —  2  ?/  =  2,    we  have 

14x  =  14,        x  =  l, 

=  28,         y  =  2. 


2 

-3 

-4  -3 

6 

-2 

X  = 

2  —2 

142/  = 

2   -4 
6       2 

EXERCISES 
Solve  by  determinants  the  systems  of  equations 
1.   8x-?/  =  34,  2.   Sx  +  iy  =  10, 

x  +  82/  =  53.  4x  +  y     =9. 


3.  «.c  +  ^i/  =  a^ 

bx  —  ay  =  ab. 


4.   Verify  that,  if  the  determinant  (2)  is  not  zero,  the  values  of  x  and  y  deter- 
mined by  division  from  (3)  satisfy  equations  (1). 

2.   Consider  a  system  of  three  linear  equations 

QiX  +  biy  +  Ciz  =  ki, 

(4)  023:  +  bojj  +  c^z  =  ki, 

CisX  +  fcg^  +  CzZ  =  ks. 

Multiply  the  members  of  the  first,  second  and  third  equations  by  * 

(5)  62C3  —  hsCo,         63C1  —  &1C3,    61C2  —  62C1, 

respectively  and  add  the  resulting  equations.  We  obtain  an  equation 
in  which  the  coefficients  of  y  and  z  are  found  to  be  zero,  while  the  coeffi- 
cient of  X  is 

(6)  ai&2C3  —  aih^C2  +  a^b^Ci  —  a-ybic-i  +  ad>iC-z  —  a^h'^Cu 

Such  an  expression  is  called  a  determinant  of  the  third  order  and  denoted 
by  the  symbol 


(6') 


The  nine  numbers  Oi,  .  .  .  ,  C3  are  called  the  elements  of  the  determi- 
nant. In  the  symbol  these  elements  lie  in  three  (horizontal)  rows,  and 
also  in  three  (vertical)  columns.  Thus  a-i,  62,  c^  are  the  elements  of  the 
second  row,  while  the  three  c's  are  the  elements  of  the  third  column. 

*  A  simple  rule  for  finding  these  multipliers  is  given  in  §  3. 


fll  &1 

Ci 

02    62 

Ci 

as  63 

Cz 

DETERMINANTS 


129 


The  equation  (free  of  y  and  z),  obtained  above,  is 


Gi   61   Ci 

/Cl   61   Ci 

02   62   C2 

a;  = 

/v2    62    C2 

as  &3  C3 

^-3    &3    Cg 

since  the  constant  member  was  the  sum  of  the  products  of  the  expres- 
sions (5)  by  ki,  ki,  ks,  and  hence  may  be  derived  from  (6)  by  replacing 
the  a's  by  the  A;'s.  Thus  the  theorem  of  §  1  holds  here  as  regards  the 
value  of  X. 

3.  Minors.  The  determinant  of  the  second  order  obtained  by  eras- 
ing (or  covering  up)  the  row  and  column  crossing  at  a  given  element  of  a 
determinant  of  the  third  order  is  called  the  minor  of  that  element.  For 
example,  in  the  determinant  D  given  by  (6'),  the  minors  of  ai,  ao,  az  are 


A, 


62  C2 
hz  Cz 


A^  = 


hi  Ci 
h  Cz 


&2    C2 


respectively.     The  multipliers  (5)  are  therefore  Ai,  —A2,  Az.     Hence  the 
first  results  obtained  in  §  2  may  be  stated  as  follows : 

(7)  D  =  aiAi  -  a^A-i  +  azAz, 

(8)  6iAi  -  62A2  +  Ms  =  0,        CiAi  -  C2A2  +  C3^3  =  0. 
The  minors  of  61,  &2,  &3  in  this  determinant  D  are 

-Si  =  a2C3  —  a3C2,         B2  =  ttiCz  —  azCi,         Bz  =  a^c-z  —  aoCi. 
Multiply  the  members  of  the  equations  (4)  by  — -Bi,  B2,  —Bz,  respectively, 
and  add.     In  the  resulting  equation,  the  coefficients  of  x  and  z  are  seen  to 
equal  zero: 

(9)  -ayBi  +  a^B.  -  azBz  =  0,         -CiBi  +  CoBo  -  CzBz  =  0, 
while  the  coefficient  of  y  is  seen  to  equal  the  expression  (6) : 

(10)  D  =  -hBi  +  62S2  -  bzBz. 
Hence  the  theorem  of  §  I  holds  here  for  the  variable  y. 

The  reader  should  also  verify  that,  if  he  uses  the  multipliers  Ci,  —C2,  Cz, 
where  C,  is  the  minor  of  Ci  in  D,  he  obtains  an  equation  in  which  the  co- 
efficients of  X  and  y  are  zero : 

(11)  aiCi  -  a2C2  +  azCz  =  0,        hd  -  62C2  +  63C3  =  0, 
while  the  coefficient  of  z  equals  the  expression  (6) : 

(12)  D  =  CiCi  -  C2C2  +  C3C3, 

and  then  conclude  that  the  theorem  of  §  1  is  true  as  regards  z. 


130 


THEORY  OF  EQUATIONS 


[Ch.  XI 


4.  Expansion  According  to  the  Elements  of  a  Column.  Relations  (7), 
(10),  (12)  are  expressed  in  words  by  saying  that  a  determinant  of  the  third 
order  may  be  expanded  according  to  the  elements  of  any  column.  To  obtain 
the  expansion,  we  multiply  each  element  of  the  column  by  the  minor  of 
the  element,  prefix  the  proper  sign  to  the  products,  and  add  the  signed 
products.     The  signs  are  alternately  +  and  — ,  as  in  the  diagram. 

+  -  + 
-  +  - 
+     -     + 

5.  Two  Columns  Alike.  A  determinant  *  is  zero  if  any  two  of  its 
columns  are  alike. 

This  is  evident  for  a  determinant  of  the  second  order: 

c  c 


dd 


=  cd  —  cd  =  0. 


To  prove  it  for  a  determinant  of  the  third  order,  we  have  only  to  expand 
it  according  to  the  elements  of  the  column  not  one  of  the  like  columns 
and  to  note  that  each  minor  is  zero,  being  a  determinant  of  the  second 
order  \vith  two  columns  alike. 

EXERCISES 

Solve  by  determinants  the  systems  of  equations  (expanding  a  determinant  having 
two  zeros  in  a  column  according  to  the  elements  of  that  column) : 

1.      x+     )j-\-    z  =  n,  2.   X  +    y  +     z  =  0, 

2X-&IJ-    z  =  0,  x  +  2y  +  3z=  -1, 

Sx  +  4y  +  2z  =  0.  x  +  Sy  -\-Gz  =  0. 

3.  Noting  that  ^i,  .42,  As  of  §  3  do  not  involve  ai,  ^2,  a^,  we  may  obtain  the 
first  expression  (S)  from  (7)  by  replacing  each  ni  by  6,-,  and  the  second  expression  (8) 
from  (7)  by  replacing  each  Ui  by  Cj.     Hence  (8)  are  the  expansions  of 


=  0 


according  to  the  elements  of  the  first  column. 

4.   Prove  similarly  that  (9)  and  (11)  fohow  from  §  o. 

*  Here  and  in  §§  6-11  we  understand  by  a  determinant  one  of  the  second  or  third 
order.  After  determinants  of  higher  orders  have  been  defined,  it  will  be  shown  that 
these  theorems  are  true  of  determinants  of  any  order. 


Ih 

Ih 

Cl 

Cl 

h 

Cl 

bo 

62 

("2 

=  0, 

Cl 

b2 

C2 

bs 

63 

C3 

Cz 

63 

C3 

§6,  7] 


DETERMINANTS 


131 


6.  Theorem.  A  determinant  having  ai  +  gi,  a^  -\-  qi,  .  .  .  as  the  ele- 
ments of  a  column  equals  the  su7n  of  the  determinant  having  ai,  Oo,  .  .  .  as 
the  elertients  of  the  corres'ponding  column  and  the  determinant  having  qi,  q^, 
.  .  .  as  the  elenierits  of  that  column,  while  the  elements  of  the  remaining 
columns  of  each  determinant  are  the  same  as  in  the  given  determinant. 

For  determinants  of  the  second  order,  there  are  only  two  cases: 


ai  +  gi  6i 
a2  +  g2  &2 

= 

tti  hi 
a2  h-2, 

+ 

qi  hi 
q2  hi 

hi  tti  +  qi 
bo  02  +  q2 

= 

hi  ai 
62  ao. 

+ 

hi  qi 
hi  qo 

For  determinants  of  the  third  order,  one  of  the  three  oases  is 
«!  +  gi  hi  ci 

02  +  g2    ho    C2 

az  +  qi  63  C3 


ai  hi  Ci 

gi  hi  ci 

= 

ao  bo  Co 

+ 

qo.  62  C2 

03    &3    C3 

g3  bz  Cz 

To  prove  the  theorem  we  have  only  to  expand  the  three  determinants 
according  to  the  elements  of  the  column  in  question  (the  first  column  in 
the  first  and  third  illustrations,  the  second  column  in  the  second  illustra- 
tion) and  note  that  the  minors  are  the  same  for  all  three  determinants. 
Hence  ai  +  gi  is  multiplied  by  the  same  minor  that  ai  and  gi  are  multi- 
plied by  separately,  and  similarly  for  a^  -\-  qo,  etc. 


7.  Removal  of  Factors.  A  common  factor  of  all  of  the  elements  of  the 
same  column  of  a  determinant  may  he  divided  out  of  the  elements  and  placed 
as  a  factor  before  the  new  determinant. 

In  other  words,  if  all  of  the  elements  of  a  column  are  divided  by  n,  the 
value  of  the  determinant  is  divided  by  n.     For  example. 


nai  hi 

=  n 

«!    &1 

noo  60 

a2  62 

ai  nhi  Ci 

ai  61  Ci 

a-i  nhi  Co 

=  n 

ao  ho  Co 

ai  nhz  Cz 

az  hz  Cz 

Proof  is  made  by  expanding  the  determinants  according  to  the  elements 
of  the  column  in  question. 


132 


THEORY  OF  EQUATIONS 


[Ch.  XI 


8.  Theorem.  A  determinant  is  not  changed  in  value  if  we  add  to  the 
elements  of  any  column  the  products  of  the  corresponding  ele7nents  of  another 
column  by  the  same  number. 


For  example,  ai  +  ^'^i  &i 

02  +  nb2  bi 
as  follows  from  the  first  relation  in  §  6 


tti  6i 
a2  bo 
Similarly,  by  the  third, 


fel 

6i 

Ci 

h 

bo 

Co 

fo3 

h^ 

Cz 

Oi  +  nbi  bi  Ci  Oi  bx  Ci 

ao  +  nbo  62  Co     =     02  62  C2     +  n 

03  +  ri&3  bs  Ci  03  63  C3 

in  which  the  last  determinant  is  zero  by  §  5. 

In  general,  let  Oi,  02,  .  .  .  be  the  elements  to  which  we  add  the  products 
of  the  elements  61,  bo,  .  .  .  by  n.  We  apply  §  6  with  gi  =  nbi,  52  =  w&2,  .... 
Thus  the  modified  determinant  equals  the  sum  of  the  initial  determinant 
and  a  determinant  having  bi,  62,  •  •  •  in  one  column  and  nbi,  nbo,  ...  in 
another  column.  But  the  latter  determinant  equals  (§7)  the  product 
of  n  by  a  determinant  with  two  columns  alike  and  hence  is  zero  (§5). 

ExL\MPLE.  Multiplying  the  elements  of  the  last  column  by  2  and  adding  the 
products  to  the  elements  of  the  second  column,  we  get 


=  -44. 


For  the  next  st(;p,  we  have  nmltiplied  the  elements  of  the  third  column  hj'  —1 
and  added  the  products  to  the  elements  of  the  first  column.  Expanding  the  third 
determinant  according  to  the  elements  of  the  third  column,  we  note  that  two  of 
the  minors  are  zero  (having  a  row  of  zeros) ,  and  hence  obtain  the  determinant  of 
the  second  order  written  above.    The  last  step  is  simplified  l^y  use  of  §  10. 

9.  Interchange  of  Rows  and  Columns.  A  determinant  is  not  altered 
if  in  its  symbol  we  take  as  the  elements  of  the  first,  second,  .  .  .  rows  the 
elements  (in  the  same  order)  which  formerly  appeared  in  the  first,  second,  .  .  . 
columns: 


1 

-2     1 

1 

0     1 

0 

0     1 

-2 

8 

1 

2    3 

= 

1 

8    3 

= 

_2 

8    3 

= 

3 

10 

6 

4     3 

6 

10    3 

3 

10     3 

D^ 


tti  61 

02  &2 

Oi  bi  Ci 

Oo    62  C2 

03    &3  C3 


Ol  O2 
61  &2 
Ol  02  O3 
by  bo  63 
Ci  C2  C3 


=  A. 


§  10,   111 


DETERMINANTS 


133 


bi  63 

ai  03 

fli  as 

+  &2 

-  C2 

61  63 

Ci   Cs 

Ci    C-i 

The  proof  is  evident  by  inspection  for  the  case  of  determinants  of  the 
second  order.  For  those  of  the  third  order,  we  expand  A  and  find  that  its 
six  terms  are  those  in  the  expansion  (6)  of  D. 

10.  Expansion  According  to  the  Elements  of  a  Row.  To  prove  that 
determinant  D,  given  by  (6'),  may  be  expanded  according  to  the  elements  of 
any  row  (say  the  second  *) : 

D  =  —a^Az  +  62^2  —  C2C2, 
with  the  same  rule  of  signs  as  in  §  4,  we  note  that  (§9) 


D  =  A=  —02 


since  A  can  be  expanded  according  to  the  elements  of  its  second  column. 
After  interchanging  the  rows  and  columns  in  these  three  determinants  of 
the  second  order,  we  have  the  minors  A2,  B2,  Co  of  02,  bo,  Co  in  D. 

Example.  The  third  determinant  in  the  Example  of  §  8  is  best  evaluated  by 
expanding  it  according  to  the  elements  of  its  first  row,  since  two  of  its  elements  are 
zero.     Indeed,  we  obtain  +1  multipHed  by  its  minor. 

11.  Theorem.  A  determinant  is  not  changed  in  value  if  we  add  to  the 
elements  of  any  row  the  products  of  the  corresponding  elements  of  another 
row  by  the  same  number. 

We  shall  show  that  D,  given  by  (6'),  equals 

ai  &i  Ci 

D'=    ao  +  nai        62  +  nbi        c-i  +  ncy 
a.3  &3  C3 

Now  D  =  A,  where  A  is  given  in  §  9.     By  §  8, 

ai    02  +  nai    as 

A  =    61     62  +  nbi     63 

Ci     Co  +  nci     C3 

Interchanging  the  rows  and  columns  of  A,  we  get  D'.     Hence 

D'  =  A  =  2). 


*  While  for  concreteness  we  have  here  (and  in  §  11)  treated  but  one  of  several  cases, 
the  proof  is  such  that  it  applies  to  all  the  cases. 


134  THEORY  OF  EQUATIONS  ICh.  xi 

EXERCISES 

1.  Evaluate  the  numerical  determinant  in  §  8  by  removing  the  factor  2  from 
the  second  column  and  then  getting  a  determinant  with  two  zeros  in  the  second 
row. 

Solve  the  systems  of  equations  (by  removing,  if  possible,  integral  factors  from  a 
column  and  reducing  each  determinant  to  one  with  two  zeros  in  a  row  before 
expanding  it) : 

2.  x-2y+     z=12,  3.   3  x  -  2  y  =  7, 
:r  +  2  iy  +  3  2  =  48,  3y-2z  =  G, 

6a;  +  47/  +  3z  =  84.  32-2.c=-l. 

Factor  a  single  determinant,  and  solve 

4.      X+     y  +    z  =  1,  5.    ax  -{-  hy  -{-  cz  =  k, 

ax-\-  hy  -\-  cz  =  k,  a-x  +  h~y  -\-  ch  =  k^, 

a-x  +  b-y  +  0^2  =  k'\  aiv  +  ¥y  +  c^2  =  k*. 

6.  Obtain  in  its  simplest  form  the  value  of  x  from 

ax  -{■    y  -\-    z  =  a  —  3, 

x-\-  ay  +    z  =  -2, 
X  -\-    y  -\-  az  =  —2. 

7.  Deduce  the  case  n  =  2  of  §  7  at  once  from  §  6,  by  taking  qi  =  a,-. 

8.  Give  the  proof  in  §  10  when  the  third  row  is  used. 

9.  Give  the  proof  in  §  11  for  a  new^  case. 

10.  A  determinant  of  the  third  order  is  zero  if  two  rows  are  alike. 

11.  Hence  prove  that  Z)'  =  Z)  in  §  11  by  expancUng  D'  according  to  the  elements 
of  its  second  row. 

12.  Prove  the  theorem  about  rows  corresponding  to  that  in  §  6. 

13.  From  Ex.  12  deduce  Ex.  11. 

12.  Definition  of  a  Determinant  of  Order  ii.  In  the  six  terms  of  the 
expression  (6),  which  was  defined  to  be  the  general  determinant  of  order  3, 
the  letters  a,  h,  c  were  always  written  in  this  sequence,  while  the  sub- 
scripts are  the  six  possible  arrangements  of  the  numbers  1,  2,  3.  The  first 
term  ai&2C3  shall  be  called  the  diagonal  term*  since  it  is  the  product  of  the 
elements  in  the  main  diagonal  running  from  the  upper  left  hand  corner  to 
the  lower  right  hand  corner  of  the  symbol  for  the  determinant.  The 
subscripts  in  the  term  —aibzCo  are  derived  from  those  of  the  diagonal 
term  by  interchanging  2  and  3,  and  the  minus  sign  is  to  be  associated 
with  the  fact  that  an  odd  number  (here  one)  of  interchanges  of  subscripts 
were  used.  To  obtain  the  arrangement  2,  3,  1  of  the  subscripts  in  the 
*  Sometimes  called  the  leading  term. 


§  12]  DETERMINANTS  135 

term  -\-a2hzC1  from  the  natural  order  1,  2,  3  (in  the  diagonal  term),  we 
may  first  interchange  1  and  2,  obtaining  2,  ^1,  3  and  then  interchange 
1  and  3;  an  even  number  (two)  of  interchanges  of  subscripts  were  used 
and  the  sign  of  the  term  is  plus. 

EXERCISES 

1.  Show  that  a  like  result  holds  for  the  last  three  terms  of  (6). 

2.  Discuss  similarly  the  two  terms  of  a  determinant  of  order  2. 

While  the  arrangement  1,  3,  2  was  obtained  from  1,  2,  3  by  one  inter- 
change (2,  3),  we  may  obtain  it  by  applying  in  succession  the  three  inter- 
changes (1,  2),  (1,  3),  (1,  2),  and  in  many  new  ways.  To  show  that  the 
number  of  interchanges  which  will  produce  the  final  arrangement  1,  3,  2 
is  odd  in  every  case,  note  that  any  interchange  (the  possible  ones  being 
the  three  just  listed)  changes  the  sign  of  the  product 

P   =    {Xi-  X2)(Xi  -  Xz){X2  -  Xi), 

where  the  a:'s  are  arbitrary  variables.  Thus  a  succession  of  k  interchanges 
yields  P  or  —P  according  as  k  is  even  or  odd.  Starting  with  the  arrange- 
ment 1,  2,  3  and  applying  k  successive  interchanges,  suppose  that  we 
obtain  the  final  arrangement  1,  3,  2.  But  if  in  P  we  replace  the  subscripts 
1,  2,  3  by  1,  3,  2,  respectively,  i.e.,  if  we  interchange  2  and  3,  we  obtain 
—  P.     Hence  k  is  odd. 

Consider  the  corresponding  question  for  n  variables.  Form  the  prod- 
uct of  all  of  the  differences  Xi  —  Xj  {i  <  j)  of  the  variables: 

P  =   (Xi-  XojiXi  -  Xs)    .    .    .   {Xi  -  Xn) 
•  (.X2  —  X3)    .    .    .    (X2  —  Xn) 


'   \Xn~l  Xn)- 

Interchange  any  two  subscripts  i  and  j.  The  factors  which  involve  neither 
i  nor  j  are  unaltered.  The  factor  di(xi  —  Xj)  involving  both  is  changed 
in  sign.     The  remaining  factors  may  be  paired  to  form  the  products 

zt{Xi  -  Xk)(xj  -  X,,)         {k  ^  1,  .  .  .  ,  n;  k  9^  i,  k  9^  j). 

Such  a  product  is  unaltered.     Hence  P  is  changed  in  sign. 

Suppose  that  an  arrangement  ii,  U,  .  .  .  ,  in  can  be  obtained  from 
1,  2,  .  .  .  ,  n  by  using  a  successive  interchanges  and  also  by  6  successive 
interchanges.     Make   these   interchanges   on   the   subscripts   in  P;    the 


136 


THEORY  OF  EQUATIONS 


[Ch.  XI 


resulting  functions  equal  {  —  lyP  and  (  — 1)^P,  respectively.  But  the 
resulting  functions  are  identical  since  either  can  be  obtained  at  one  step 
from  P  by  replacing  the  subscript  1  by  d,  2  by  12,  .  .  .  ,  n  by  i^.     Hence 

i-iyp^{-iyp, 

so  that  a  and  b  are  both  even  or  both  odd. 
We  define  a  determinant  of  order  4  to  be 


(13) 


«!  bi  Ci  di 

a2  62  C2  di 

as  63  Cz  dz 

cii  bi  Ci  di 


2  ±  aqbrCA, 


(24) 


where  q,  r,  s,  t  is  any  one  of  the  24  arrangements  of  1,  2,  3,  4,  and  the 
sign  of  the  corresponding  term  is  +  or  —  according  as  an  even  or  odd 
number  of  interchanges  are  needed  to  derive  this  arrangement  q,  r,  s,  t 
from  1,  2,  3,  4.  Although  different  numbers  of  interchanges  will  produce 
the  same  arrangement  q,  r,  s,  t  from  1,  2,  3,  4,  these  numbers  are  all  even 
or  all  odd,  as  just  proved,  so  that  the  sign  is  fully  determined. 

We  have  seen  that  the  analogous  definitions  of  determinants  of  orders 
2  and  3  lead  to  our  earUer  expressions  (2)  and  (6). 

We  will  have  no  difficulty  in  extending  the  definition  to  a  determinant 
of  general  order  n  as  soon  as  we  decide  upon  a  proper  notation  for  the  ti^ 
elements.  The  subscripts  1,  2,  .  .  .  ,  n  may  be  used  as  before  to  specify 
the  rows.  But  the  alphabet  does  not  contain  n  letters  with  which  to 
specify  the  columns.  The  use  of  e',  e",  .  .  .  ,  e^"^  for  this  purpose  would 
conflict  Avith  the  notation  for  derivatives  and  l)esides  be  very  awkward 
when  exponents  are  used.  It  is  customary  in  mathematical  journals  and 
scientific  books  (a  custom  not  always  followed  in  introductory  text  books, 
to  the  distinct  disadvantage  of  the  reader)  to  denote  the  n  letters  used  to 
distinguish  the  /)  columns  by  d,  €2,  .  .  .  ,  €„  (or  some  other  letter  with 
the  same  subscripts)  and  to  prefix  (but  see  §  13)  such  a  subscript  by 
the  subscript  indicating  the  row.     The  symbol  for  the   determinant  is 

therefore 

en  ^12...  ein 


(14) 


D  = 


C-21     e-22 


€nl  en2 


e2n 


e„„ 


§  131  DETERMINANTS  137 

Bj^  definition  *  this  shall  mean  the  sum  of  the  n!  terms 

(14')  (-iyc,,ic,,2  .  .  .ei^^n 

in  which  ii,  ii,  ■  ■  -  ,  in  is  an  arrangement  of  1,  2,  .  .  .  ,  n,  derived  from 
I,  2,  .  .  .  ,  71  hy  i  interchanges.  For  example,  if  we  take  n  =  4  and 
write  ay,  6y,  Cy,  dj  for  eyi,  ey2,  ey3,  Cji,  the  symbol  (14)  becomes  (13)  and  the 
general  term  (14')  becomes  (—  1)'  a,,  hi^  Ci^  d,-,,  the  general  term  of  the  second 
member  of  (13). 

EXERCISES 

1.  Give  the  six  terms  involving  (h  in  the  determinant  (13). 

2.  What  are  the  signs  of  aJb^C'dxeA,  ihhiCzd^ei  in  a  determinant  of  order  five? 

3.  The  arrangement  4,  1,  3,  2  may  be  obtained  from  1,  2,  3,  4  by  use  of  the  two 
successive  interchanges  (1,  4),  (1,  2),  and  also  by  use  of  the  four  successive  inter- 
changes (1,  4),  (1,  3),  (1,2),  (2,3). 

4.  Write  out  the  six  terms  of  (14)  for  n  —  3,  rearrange  the  factors  of  each  term 
so  that  the  new  first  subscripts  shall  be  in  the  order  1,  2,  3,  and  verify  that  the 
resulting  six  terms  are  those  of  the  expansion  of  U  in  §  13  for  n  =  3. 

13.   Interchange  of  Rows  and  Columns.     Determinant  (14)  equals 


D'  = 


en  en   .  .  .  e„i 

6l2     C22     •     •     •    6„2 
e\n  e2n    •     •     •    enn 


Without  altering  (14'),  we  may  rearrange  its  factors  so  that  the  first 
subscripts  shall  appear  in  the  order  1,  2,  .  .  .  ,  n,  and  get 

{  —  I)'eik,e2k2  ■  •  •  enk„- 

Since  this  can  be  done  by  i  interchanges  of  the  letters  e  (corresponding  to 
the  i  interchanges  by  which  the  first  subscripts  ii,  .  .  .  ,  in  were  derived 
from  1,  .  .  .  ,  n),  the  new  second  subscripts  ki,  .  .  .  ,  kn  are  derived  from 
the  old  second  subscripts  I,  .  .  .  ,  n  by  i  interchanges.  The  resulting 
signed  product  is  therefore  a  term  of  D'.     Hence  D  =  D'. 

*  We  may  define  a  determinant  of  order  n  by  mathematical  induction  from  n  —  1 
to  n,  using  the  first  equation  in  §  17.  The  next  step  would  be  to  prove  that  the  present 
definition  holds  as  a  theorem. ' 


138  THEORY  OF  EQUATIONS  [Ch.  XI 

14.  Interchange  of  Two  Columns.  A  determinant  is  changed  in.  sign 
by  the  interchange  of  any  two  of  its:  coliinms. 

Let  A  be  the  determiniint  derived  from  (14)  b}-  the  interchange  of  the 
rth  and  sth  columns.  The  expansion  of  A  is  therefore  obtained  from 
that  of  D  by  interchanging  r  and  6-  in  the  series  of  second  subscripts  of 
each  term  (14')  of  D.  Interchange  the  rth  and  sth  letters  e  to  restore 
the  second  subscripts  to  their  natural  order.  Since  the  first  subscripts 
have  undergone  an  interchange,  the  negative  of  any  term  of  A  is  a  term 
of  D,  and  A  =-D. 

15.  Interchange  of  Two  Rows.  A  determinant  D  is  cha7iged  in  sign 
by  the  interchange  of  any  two  rows. 

Let  A  be  the  determinant  obtained  from  D  by  interchanging  the  rth 
and  sth  rows.  By  interchanging  the  rows  and  columns  in  D  and  in  A ,  we 
get  two  determinants  D'  and  A',  either  of  which  may  be  derived  from  the 
other  by  the  interchange  of  the  rth  and  sth  columns.     Hence,  by  §§  13,  14, 

A  =  A'=  -D'  =  -D. 

16.  Two  Rows  or  Two  Columns  Alike.  A  determinant  is  zerojf  any 
two  of  its  rows  or  any  two  of  its  columns  are  alike. 

For,  by  the  interchange  of  the  two  like  rows  or  two  like  columns,  the 
determinant  is  evidently  unaltered,  and  yet  must  change  in  sign  by  §§  14. 
15.     Hence  D  =  -D,  D  =  0. 

17.  Expansion.  A  determinant  can  he  expanded  according  to  the  ele- 
ments of  any  row  or  any  column. 

Let  Eij  be  the  minor  of  e,/  in  D,  given  by  (14).  Thus  E',y  is  the  deter- 
minant of  order  n  —  I  obtained  by  erasing  the  ith  row  and  the  jth  column 
(crossing  at  Cij).     We  first  prove  that 

D^CnEn-eoiEn  +  e.iEn-   •  ■  ■  +  (-l)"-'e„i£J„i, 

so  that  D  can  be  expanded  according  to  the  elements  of  its  first  column. 
The  terms  of  D  with  the  factor  en  are  of  the  form 

(-l)*eiie,-,2  .  .  .  e,„n, 

where  i2,  .  .  .  ,  i'„  is  an  arrangement  oi  2,  .  .  .  ,  n  derived  from  the  latter 
by  i  interchanges.  Removing  from  each  term  the  factor  en,  antl  adding 
the  quotients,  we  obtain  tlie  {n  —  1)!  properly  signed  terms  of  En- 


§  181 


DETERMINANTS 


139 


Let  A  be  the  determinant  obtained  from  D  by  interchanging  the  first 
and  second  rows.  As  just  proved,  the  total  coefficient  of  621  in  A  is  the 
minor 

ei2  ei3    ...  ein 

C32     633       ...    Csn 


en2  era 


Cnr, 


of  621  in  A.  Now  this  minor  is  identical  with  E21.  But  A  =  —D  (§  15). 
Hence  the  total  coefficient  of  e-^  in  D  equals  —£'21. 

Similarly,  the  coefficient  of  631  is  -£"31,  etc. 

To  obtain  the  expansion  of  D  according  to  the  elements  of  its  kth  col- 
umn, where  k  >  1,  we  consider  the  determinant  5  derived  from  D  by 
moving  the  kth  column  over  the  earlier  columns  until  it  becomes  the  new 
first  column. 

Since  this  may  be  done  by  A;  —  1  interchanges  of  adjacent  columns, 
8  =  (— l)*^~iZ).  The  minors  of  the  elements  en,,  .  .  .  ,  e„fc  in  the  first 
column  of  8  are  evidently  the  minors  Eik,  .  .  .  ,  Enk  of  ei^,  .  .  .  ,  €«&  in  D. 
Hence,  by  the  earlier  result, 


(15) 


{k  =  1,  .  .  .  ,n). 


y=i 


Applying  this  result  to  the  equal  determinant  D'  of  §  13,  and  changing 
the  summation  index  from  j  to  k,  we  get 


(16) 


(i  =  1,  .  .  .  ,  n). 


jt=i 


This  gives  the  expansion  of  D  according  to  the  elements  of  the  jth  row. 
One  decided  advantage  of  the  double  subscript  notation  is  the  resulting 
simplicity  of  the  last  two  expansions.  Of  course  the  sign  may  also  be 
found  by  counting  spaces  as  in  §  4. 


18.  The  theorems  in  §§  6-8,  11  now  follow  for  determinants  of  order  n. 
Indeed,  the  proofs  were  so  worded  that  they  now  apply,  since  the  auxiliary 
theorems  used  have  been  extended  (§§  13,  16,  17)  to  determinants  of 
order  n. 


140 


THEORY  OF  EQUATIONS 


(Ch.  XI 


EXERCISES 
1.   Prove  the  theorem  of  §  15  b}'  the  direct  method  of  §  14. 
b  -{-  c      c  -\-  a      a  -{-  b 
bi  +  Ci     Ci  +  (h     ai  +  bi 

62  +  C2       f  2  +  02       02  +  62 

By  reducing  to  a  determinant  of  order  3,  etc.,  prove  that 


a 

b 

c 

«! 

h 

Ci 

02 

b. 

C2 

2 

-1 

3 

-2 

1 

7      1    -1 

A' 

3 

5-5       3 

—    — ^i 

4 

-3      2    -1 

b 

c     d 

b' 

C2       d^ 
C3      d^ 

=  abcd(a  —  b) 

b' 

c*     d^ 

ae  +  bg    af  +  bh 

ce 

+  dg 

cf  +  dh 

4. 


1     1 
1     2 


1 

1 

3 

4 

6 

10 

10 

20 

=  1. 


abcdia  —  b){a  —  c){a  —  d){b  —  c){b  —  d){c  —  d). 


a    b 

e    f 

c    d 

g    h 

[use  §  6]. 


aici  +  bid  +  CiCz  aji  +  h/n  +  Ctfs  oigi  +  big2  +  Ci^s 
0261  +  &2S2  +  0263  02/1  +  62/2  4-  C'fs  a-igi  +  62^2  +  Cog 3 
a-iCi  4-  6362  +  CsCs     aa/i  +  63/2  +  C3/3     Osgri  +  63(72  +  ^3^3 


ai 

61 

Ci 

02 

62 

C2 

• 

03 

^3 

C3 

Ci 

/i 

fifi 

02 

/2 

!72 

es 

/a 

f73 

Write  out  only  the  6  of  the  27  determinants  (§  6)  which  are  not  necessarily  zero. 

8.  Hence  verify  that  the  product  of  two  determinants  of  the  same  order  (2  or  3) 
is  a  determinant  of  like  order  in  which  the  element  of  the  rth  row  and  cth  column 
is  the  sum  of  the  products  of  the  elements  of  the  /Ih  row  of  the  first  determinant 
by  the  correspond in^j;  elements  of  the  cth  column  of  the  second. 

9.  Express  (a^  +  6^  +  c^  +  d-){e-  4"/"  +  ^"  +  h'')  as  a  smn  of  4  squares  by 
writing 

e  +  fi    g  +  hi 


a-\-bi     c-{-  di 
—  c-\-di    a  —  bi 


■g  +  hi    e  —  fi 


as  a  determinant  of  order  2  similar  to  eacli  factor. 


§'19,  201 


DETERMINANTS 


141 


10.    If  Si  =  a'  +  ^'  +  tS 

1        1        1 


3 

Si 

S2 

Sl 

S2 

S3 

S2 

S3 

S4 

11.  Using  the  Factor  Theorem  and  the  diagonal  term,  prove  Ex.  5  and 


1        1 

Xi         Xi 
X\        X2" 


1 


Xn 


»  n(n-l) 


where  P  is  given  in  §  12. 

12.   With  the  notations  of  §  .3,  and  using  (7)-(12),  prove  that 


A, 

-A2 

A3 

B, 

B, 

-Bz 

• 

c\ 

-C2 

Cz 

O]      61      Ci 

a^    &2    C2 

03       63       Cz 


\D 

0   0 

=     0 

D   0 

;  0 

0   D 

Hence  the  first  determinant  equals  D-. 

19.    Complementary  Minors.     The  determinant  D  of  order  4  in  (13) 
is  said  to  have  the  two-rowed  complementary  minors 


M  = 


az  &3 


M'  = 


C2  (h 
Ci  di 


since  either  is  obtained  by  erasing  from  D  all  the  rows  and  columns  having 
an  element  occurring  in  the  other.  Similarly,  any  r-rowed  minor  of  a 
determinant  of  order  n  has  a  definite  complementary-  (n  —  r)-rowed 
minor.  In  particular,  any  element  is  regarded  as  a  one-rowed  minor 
and  is  complementary  to  its  minor. 

20.  Laplace's  Development.  Any  determinant  D  equals  the  sum  of 
all  the  products  ±  MM',  where  M  is  an  r-rowed  minor  having  its  elements 
in  the  first  r  columns  of  D,  and  M'  is  the  minor  complementary  to  M,  while 
the  sign  is  -{-  or  —  according  as  an  even  or  odd  number  of  interchanges  of 
rows  of  D  will  bring  M  into  the  position  occupied  by  the  minor  Mi  whose 
elements  lie  in  the  first  r  rows  and  first  r  columns  of  D. 


142 


THEORY  OF  EQUATIONS 


ICh.  XI 


For  r  =  1,  this  development  becomes  the  known  expansion  of  D  according  to 
the  elements  of  the  first  column  (§  17);  here  Mi  =  Cu. 
If  r  =  2  and  D  is  the  determinant  (13)  of  order  4, 


Oi 

6i 

Ca 

dz 

Oi 

61 

Ci 

rf2 

( 

Oi 

61 

C2 

d2 

D  = 

. 

• 

+ 

. 

02 

h 

C4 

di 

03 

63 

Ci 

d. 

04 

64 

C3 

dz 

02 

&2      Cl 

di 

02 

62 

Cl 

rfl 

03 

&3 

Cl 

di 

+ 

— 

• 

+ 

03 

63 

C4 

d, 

04 

&4 

C3 

C^3 

04 

64 

1 

C2 

d^ 

The  first  term  of  the  development  is  MiJ//;  the  second  term  is  — il/.l/'(in  the  nota- 
tions of  §  19),  and  the  sign  is  minus  since  the  interchange  of  the  second  and  third 
rows  of  D  brings  this  M  into  the  position  of  Mi.  The  sign  of  the  third  term  of  the 
development  is  plus  since  two  interchanges  of  rows  of  D  bring  the  first  factor 
into  the  position  of  My. 


If  D  is  the  determinant  (14),  then 


Ml  = 


en 


en 


eir 


err 


Ml'  = 


Cr+lr+l    • 


er+\, 


en  r+\ 


6n  n 


Any  term  of  the  product  Miilf  /  is  of  the  type 


{-lyei.xe^ 


ei^r  '  \        i-J   Ci^_|.,r+1     •     .     •    ei^fi) 


where  r'l,  .  .  .  ,  ^V  is  an  arrangement  of  1,  .  .  .  ,  r  derived  from  1,  .  .  ,  ,  r 
by  I  interchanges,  while  iV+i,  •  •  •  ,  in  is  an  arrangement  of  r  +  1,  •  •  •  ,  n 
derived  by  j  interchanges.  Hence  ii,  .  .  .  ,  tn  is  an  arrangement  of 
I,  .  .  .  ,  n  derived  by  i  -jr  j  interchanges,  so  that  the  above  product  is  a 
term  of  D  with  the  proper  sign. 

It  now  follows  from  §  15  that  any  term  of  any  of  the  products  ±  MM' 
of  the  theorem  is  a  term  of  D.  Clearly  we  do  not  obtain  in  this  manner 
the  same  term  of  D  twice. 

Conversely,  any  term  t  oi  D  occurs  in  one  of  the  products  ±  MM'. 
Indeed,  t  contains  as  factors  r  elements  from  the  first  r  columns  of  D, 
no  two  being  in  the  same  row,  and  the  product  of  these  is,  except  per- 
haps as  to  sign,  a  term  of  some  minor  M.  Thus  t  is  a  term  of  MM'  or 
of  —MM'.  In  view  of  the  earlier  discussion,  the  sign  of  t  is  that  of  the 
corresponding  term  in  ±  MM',  where  the  latter  sign  is  given  by  the 
theorem. 


§  21.  22j 


DETERMINANTS 


143 


21.  There  is  a  Laplace  development  of  D  in  which  the  r-rowed  minors 
M  have  their  elements  in  the  first  r  rows  of  D,  instead  of  in  the  first  r 
columns  as  in  §  20.  To  prove  this,  we  have  only  to  apply  §  20  to  the  equal 
determinant  obtained  by  interchanging  the  rows  and  columns  of  D. 

There  are  more  general  (but  less  used)  Laplace  developments  in  which 
the  r-rowed  minors  M  have  their  elements  in  any  chosen  r  columns  (or 
rows)  of  D.  It  is  simpler  to  apply  the  earlier  developments  to  the  de- 
terminant ±  D  having  the  elements  of  the  chosen  r  columns  (or  rows) 
in  the  new  first  r  columns  (or  rows). 


EXERCISES 


L. 

a    b     c 

d 

e    f    g 

h 

a     b 

J 

k 

0    0    j 

k 

c     I    ' 

I 

m 

0    0     I 

m 

I 

a    b    c 

d 

1 
2 

e    f   g 
a    b    c 

e    f   g 

h 
d 
h 

^ 

a    b 
e    f 

c 

g 

d 
h 

- 

a    c 
e    g 

• 

b    d 

a    d 

e    h 

• 

=  0. 


3.  Check  §  20  by  showing  that  the  total  number  of  products  of  n  elements  is 
Cr"  '  r\{n  —  r)\  =  n\,  where  Cr"  is  the  number  of  combinations  of  n  things  r  at  a 
time. 

For  Laplace's  development  of  many  special  determinants,  see  Ch.  XIL 

22.  Product  of  Determinants.  The  important  rule  (Ex.  8,  p.  140), 
for  expressing  the  product  of  two  determinants  of  order  n  as  a  determi- 
nant of  order  n  is  found  and  proved  easily  by  means  of  Laplace's  develop- 
ment. For  brevity  we  shall  take  n  =  3,  but  the  method  is  seen  to  apply 
for  any  n.     We  have 


ai    6i 

Ci 

ei    /i     g\ 

a-2    hi 

C2 

. 

62    h     g2 

az    63 

Cz 

ez    fz     gz 

ai 

&i 

Ci 

0 

0 

0 

Ch 

^2 

C2 

0 

0 

0 

Ci-i 

63 

Cz 

0 

0 

0 

-1 

0 

0 

Ci 

/. 

gl 

0 

-1 

0 

62 

h 

Qi 

0 

0 

-1 

ez 

h 

gs 

144  THEORY  OF  EQUATIONS  (Ch.  xi 

In  the  determinant  of  order  6,  multiply  the  elements  of  the  first  column 
by  ei,  fi,  gi  in  turn  and  add  the  products  to  the  corresponding  elements  of 
the  fourth,  fifth  and  sixth  columns,  respectively  (and  hence  introduce 
zeros  in  place  of  the  present  elements  Ci, /i,  ^i).  Then  multiply  the  ele- 
ments of  the  second  column  by  62,  f-i,  92  in  turn  and  add  the  products  to 
the  corresponding  elements  of  the  fourth,  fifth  and  sixth  columns,  re- 
spectively. Finally,  multiply  the  elements  of  the  tliird  column  by  63,  /s,  gs 
in  turn  and  add  as  before.     The  new  determinant  is 


ai 

61 

Ci 

aiei+6ie2+cie3 

ai/i+&i/2+ci/3 

01^1+ 61^2 +C1C/3 

02 

&2 

C2 

0261  + 6262 +  C2e3 

02/1+62/2+02/3 

02^1  + 62^2 +  C2C/3 

as 

h 

C3 

0361+ 6362 +£363 

03/1+63/2+03/3 

azgi-\-hzg2+Czgz 

-1 

0 

0 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

0 

0 

-1 

0 

0 

0 

By  Laplace's  development  (or  by  expansion  according  to  the  elements  of 
the  last  row,  etc.),  this  equals  the  3-rowed  minor  whose  elements  are  the 
long  sums,  and  written  in  Ex.  7,  p.  140. 

23.   Systems  of  Linear  Equations.     l\\  the  n  equations 

OuXi  +  ai2X2  +  •  •  •  +  oi„a:„  =  ki, 
(17)  

On  1X1  +  ClniXz  +     •     •     •     +  dnnXn    =   A:„, 

let  D  denote  the  determinant  of  the  coefficients  of  the  n  unknowns : 


D  = 


Oji  ai2  .  .  .  oi, 


Onl    0„2    •     •     •    On, 

Let  Aij  be  the  minor  of  a,;  in  D.  Multiply  the  members  of  the  first  equa- 
tion by  All,  those  of  the  second  equation  by  — A21,  .  .  •  ,  those  of  the  /ith 
equation  by  (— 1)"~^  A„i,  and  add.  The  coefficient  of  Xi  is  the  expansion 
of  D  according  to  the  elements  of  its  first  column.  The  coefficient  of  Xz 
is  the  expansion,  according  to  the  elements  of  the  first  column,  of  a  deter- 
minant derived  from  D  by  replacing  an  by  an,  •  .  .  ,  o^ii  by  a„2,  so  that 
this  determinant  has  the  first  two  columns  alike  and  hence  is  zero.  In 
this  manner,  we  find  that 
(18)  DXi  =  Ki,         Dxi  =  K2,  .  .  .  ,        Dxn  =  Kn, 


5  24) 


DETERMINANTS 


145 


in  which  (see  (|3)  of  §  24)  Ki  is  derived  from  D  by  substituting  ki,  .  .  .  ,  kn 
for  the  elements  an,  .  .  .  ,  Uni  of  the  tth  column  of  D.  Another  proof  of 
(18)  follows  from 


Dxi  = 


anXi      an 

ttnlXi      a„2 


ain 

ann 


anXi  + 

an].Xi-\-  ■ 


■  +  ai  nXn 


Oi„ 


=  Kx. 


We  have  now  extended  to  any  n  results  proved  for  n  =  2  and  n  =  3 
in  §§  1-3. 

If  D  9^  0,  the  unique  values  of  a;i,  .  .  .  ,  Xn  determined  by  division 
from  (18)  actually  satisfy  equations  (17).  For  instance,  the  first  equation 
is  satisfied  since 


kiD  —  anKi  —  anKi 


ainKn  = 


ki 

an 

ai2  . 

.  ain 

h 

an 

ai2  . 

.  ai„ 

ki 

021 

^22    . 

•    02  n 

fCn 

flnl 

a„2  . 

.  a„„ 

as  shown  by  expansion  according  to  the  elements  of  the  first  row;  and 
the  determinant  is  zero,  having  two  rows  alike. 

24.  Rank  of  a  Determinant.  If  a  determinant  D  of  order  n  is  not 
zero,  it  is  said  to  be  of  rank  n.  In  general,  if  some  r-rowed  minor  of  D  is 
not  zero,  while  every  (r  +  1) -rowed  minor  is  zero,  D  is  said  to  be  of 
rank  r. 

For  example,  a  determinant  D  of  order  3  is  of  rank  3  if  Z)  f^  0;  of  rank  2  if  D  =  0, 
but  some  two-rowed  minor  is  not  zero;  it  is  of  rank  1  if  every  two-rowed  minor 
is  zero,  but  some  element  is  not  zero.  It  is  said  to  be  of  rank  0  if  every  element  is 
zero. 

In  the  discussion  of  the  three  equations  (4),  five  cases  arise: 

(a)  D  of  rank  3,  i.e.,/)  ?^  0. 

(/3)  D  of  rank  2  {i.e.,  D  =  0,  but  some  two-rowed  minor  ^  0),  and 


K,= 


ki     bi    Ci 
k2    62    C2 

^"3       bs       Ci 


K2   = 


Ol 

h 

Ci 

a2 

h 

Ci 

aa 

ki 

C3 

Ki  = 


ai 

bi 

h 

02 

62 

h 

as 

h 

h 

not  all  zero. 


146  THEORY  OF  EQUATIONS  [Ch.  XI 

(7)  D  of  rank  2  and  K\,  K->,  K3  all  zero. 

(5)  D  of  rank  1  {i.e.,  every  two-rowed  minor  =  0,  but  some  element  ^  0),  and 


{I,  j  chosen  from  1,  2,  3) 


di    ki  bi    ki  d    ki  I 

dj    kj  bj    kj  cj    kj  I 

not  all  zero;  there  are  nine  such  determinants  K. 

(«)  D  of  rank  1,  and  all  nine  of  the  determinants  K  zero. 

In  case  (a)  the  equations  have  a  single  set  of  solutions  (§  23).  In  cases  (ff)  and 
(5)  there  is  no  set  of  solutions.  In  case  (7)  one  of  the  equations  is  a  linear  com- 
bination of  the  other  two;  for  example,  if  (hb-y  —  a-ibi  9^  0,  the  first  two  equations 
determine  x  and  y  as  linear  functions  of  z  (as  shown  by  transposing  the  terms  in  z 
and  sohang  the  resulting  equations  for  x  and  y),  and  the  resulting  values  of  x  and 
y  satisfy  the  third  equation  identically  as  to  z.  Finally,  in  case  (e),  two  of  the 
equations  are  obtained  by  multiplying  the  remaining  one  by  constants.  For 
(/3)  the  proof  follows  from  (18).     For  (7),  (5),  (e),  the  proof  is  given  in  §  25. 

The  reader  acquainted  wnth  the  elements  of  solid  analytic  geometry  will  see 
that  the  planes  represented  by  the  three  equations  have  the  foUo\ving  relations: 

(a)  The  3  planes  intersect  in  a  single  point. 

(fS)  Two  of  the  planes  intersect  in  a  line  parallel  to  the  third  plane. 

(7)  The  3  planes  intersect  in  a  common  line. 

(S)   The  3  planes  are  parallel  and  not  all  coincident. 

(e)    The  3  planes  coincide. 

25.  Fundamental  Theorem.  Let  the  determinant  D  of  the  coefficients 
of  the  unknowns  in  equations  (17)  he  of  rank  r,  r  <  n.  If  the  determinants 
K  obtained  from  the  (r  +  l)-roived  minors  of  D  by  replacing  the  elements 
of  any  column  by  the  corresponding  known  terms  ki  are  not  all  zero,  the  equa- 
tions are  inconsistent.  But  if  these  determinants  K  are  all  zero,  the  r  equa- 
tions involving  the  elements  of  a  non-vanishing  r-rowed  minor  of  D  determine 
uniquely  r  of  the  variables  as  linear  functions  of  the  remaining  n  —  r  vari- 
ables, and  the  expressions  for  these  r  variables  satisfy  also  the  remaining 
n  —  r  equations. 

For  example,  letr  =  n  —\.  Then  D  =  0  and  the  K's  arc  the  Ki,  .  .  .  , 
Kn  of  §  23.  Hence,  by  (18),  the  equations  are  inconsistent  unless 
Ki,  .  .  .  ,  Kn  are  all  zero.     This  affords  an  illustration  of  the  following 

Le.m.ma  1.  If  every  (r  +  l)-rowed  minor  .1/  formed  from  certain 
r  +  1  rows  of  D  is  zero,  the  corresponding  r  -f  1  equations  (17)  are  incon- 
sistent if  there  is  a  non-vanishing  determinant  K  formed  from  any  M 
by  replacing  the  elements  of  any  column  by  the  corresponding  known 
terms  ki. 


251 


DETERMINANTS 


147 


For  concreteness,*  let  the  rows  in  question  be  the  first  r  +  1  and  let 
ctii      .  ,  .  ttif        ki 


K  = 


5^0. 


dr+l  1    .     .     .    dr+l  T  kr+i 

Let  di,  .  .  .  ,  dr+l  be  the  minors  of  ^i,  .  .  .  ,  kr+i  in  K.  Multiply  the 
first  r  +  1  equations  (17)  by  di,  —di,  .  .  .  ,  {  —  ly  dr+i,  respectively,  and 
add.  The  right  member  of  the  resulting  equation  is  ±  K.  The  coeffi- 
cient of  Xs  is 

an      ...  air         ai , 


dr+l 1    •     .     .   dr+l 

and  is  zero,  being  an  M.     Hence  0  =  ±iv. 


dr+l 


Lemma  2.  If  all  of  the  determinants  M  and  K  in  Lemma  1  are  zero, 
but  an  r-rowed  minor  of  an  M  is  not  zero,  one  of  the  corresponding  r  +  1 
equations  is  a  linear  combination  of  the  remaining  r  equations. 

As  before  let  the  r  +  1  rows  in  question  be  the  first  r  +  1.  Let  the 
non-vanishing  r-rowed  minor  be 

dii  .  .  .  ai, 


(19) 


dr+l  = 


drl 


5^0. 


Let  the  functions  obtained  by  transposing  the  terms  ki  in  (17)  be 

Li  =  diiXi  +  di2X2  +     •     •    •     +  dinXn  —  ki. 

By  the  multiplication  made  in  the  proof  of  Lemma  1, 

diLi  -  doLi -\-  ■  •  •  -\-  {-l)'dr+iLr+i  =  ^K  =  0. 
Hence  Lr+i  is  a  linear  combination  of  Li,  .  .  .  ,  Lr. 

The  first  part  of  the  fundamental  theorem  is  true  by  Lemma  1.  The 
second  part  is  readily  proved  by  means  of  Lemma  2.  Let  (19)  be  the 
non-vanishing  r-rowed  minor  of  D.  For  s  >  r,  the  .sth  equation  is  a 
linear  combination  of  the  first  r  equations,  and  hence  is  satisfied  by  any 
set  of  solutions  of  the  latter.  In  the  latter  transpose  the  terms  involving 
Xr+i,  .  .  •  ,  Xn.  Since  the  determinant  of  the  coefficients  of  rci,  .  .  .  ,  Xr 
is  not  zero,  §  23  shows  that  Xi,  .  .  .  ,  Xr  are  uniquely  determined  linear 
functions  of  Xr+i,  .  .  .  ,  x„  (which  enter  from  the  new  right  members). 

*  All  other  cases  may  be  reduced  to  this  one  by  rearranging  the  n  equations  and 
relabeUing  the  unknowns  (replacing  Xi  by  the  new  .Ci,  for  example). 


148  -^      [theory  of  equations  [Ch.  XI 

EXERCISES 

1.  Write  out  the  proof  of  the  theorem  in  §  25  for  the  cases  (7),  (5),  (e)  in  §  24. 
Discuss  the  following  systems  of  equations: 

2.  2x-\-  y  +  Sz  =  l,  3.  2x+  ?/+  32=  1, 
4:x  +  2y-  z=-S,  4.r  +  27/-  2  =  3, 
2x+    ?/-42  =  -4.  2x+     ?/-42  =  4. 

4.      X-    3y+    42  =  1,  5.     x-   3y  +    42  =  1, 

4x- 12?/ +162  =  3,  4x-- 12;/  +  162  =  4, 

3x-    9y-\-12z  =  3.  3x-    9y  +  12z  =  3. 

6.  Discuss  the  equations  in  Exs.  4  and  5,  p.  134,  when  two  or  more  of  the  num- 
bers a,  b,  c,  k  are  equal. 

7.  Discuss  the  equations  in  Ex.  6,  p.  134,  when  a  —  —2. 

26.  Homogeneous  Linear  Equations.  When  the  known  terms  A:i,  ...  , 
kn  in  (17)  are  all  zero,  the  equations  are  called  homogeneous.  The  determi- 
nants K  are  now  all  zero,  so  that  the  n  homogeneous  equations  are  never 
inconsistent.  This  is  also  evident  from  the  fact  that  they  have  the  set  of 
solutions  a;i  =  0,  .  .  .  ,  .T„  =  0.  By  (18),  there  is  no  further  set  of  solu- 
tions if  D  5^  0.  If  D  =  0,  there  are  further  sets  of  solutions:  if  D  is  of 
rank  r,  there  occur  n  —  r  arbitrary  parameters  in  the  general  set  of  solu- 
tions (§  25).     A  particular  case  of  this  result  is  the  much  used  theorem: 

A  necessary  and  sufficient  condition  that  n  linear  homogeneous  equations 
in  n  unknowns  shall  have  a  set  of  solutions,  other  than  the  trivial  one  in  which 
each  unknown  is  zero,  is  that  the  determinant  of  the  coefficients  he  zero. 

27.  The  case  of  a  sj^stem  of  fewer  than  n  linear  equations  in  n  un- 
knowns may  be  treated  by  means  of  the  Lemmas  in  §  25. 

In  case  we  have  a  system  of  more  than  n  linear  equations  in  n  unknowns, 
we  may  first  discuss  n  of  the  equations.  If  these  are  inconsistent,  the 
entire  system  is.  If  they  are  consistent,  the  general  set  S  of  solutions  may 
be  found  and  substituted  into  the  remaining  equations.  There  result 
conditions  on  the  parameters  occurring  in  »S\  and  these  linear  conditions 
may  be  treated  in  the  usual  manner.  Ultimately  we  get  either  the  gen- 
eral set  of  solutions  of  the  entire  system  of  equations  or  find  that  they  are 
inconsistent.  To  decide  in  advance  which  of  these  cases  will  arise  we  have 
only  to  find  the  maximum  order  r  of  a  non-vanishing  r-rowed  determinant 
formed  from  the  coefficients  of  the  unknowns,  taken  in  the  regular  order 


28] 


DETERMINANTS 


149 


in  which  they  occur  in  the  equations,  and  ascertain  whether  or  not  the 
(r  +  l)-rowed  determinants  K,  formed  as  in  §  25,  are  all  zero.* 

28.  An  important  case  is  that  of  n  non-homogeneous  linear  equations 
in  n  —  1  unknowns  Xi,  .  .  .  ,  Xn-u  By  multiplying  the  known  terms  by 
a:„  =  1,  we  bring  this  case  under  that  of  n  homogeneous  linear  equations 
in  n  unknowns  (§  26).  Then  (18)  gives  Dx^  =  0,  Z)  =  0,  so  that  the  given 
equations  are  inconsistent  if  D  ?^  0. 

There  is  no  set  of  solutions  of  the  n  equations 


aiio^i  +  ai2X2  + 


+  ai„-iXn-i  —  ki, 

"T"  O/n  n—lXn—l  "'/i, 


an 


ani 


dln-l    ki 
an  n— 1  l^n 


9^0. 


EXERCISES 

Discuss  the  following  systems  of  equations: 
1.   x+    y  +  3z  =  0,       2.   2x-       y+    4:2  =  0, 
x  +  2y  +  2z  =  C  x+    'iy  -    22  =  0, 

x  +  oi/-    2=0.  x-lly  +Uz  =  Q. 


4.   6  X-  +  4  ?/  +  3  2  -  84  WJ  =  0, 

x  +  2  7/  +  3  2  -  48  w  =  0, 

X  -2y  +    2  -  12  w  =  0, 

4x-  +  4?/-     2-24t/;  =  0. 

6.   2  X  +    1/  +  3  2  =  1, 
4x  +  2y    -  2  =  -3, 
2x+    y-4:z=  -4, 
10x  +  5y-Gz=  -10. 


3.  .T  -  3  ?/  +  4  2  =  0, 
4a;  -12y +  162  =  0, 
3.C  -  9y  +  12z=--0. 

2.C+  3y-  42+  5  iv  =  0, 
3 X  +  5y  —  2  +  2  li;  =  0, 
7x+ny-  92  +  12i«  =  0, 
3x+    4?/-  11  2  +  13  a;  =  0. 

2  X  -     y  +  Sz  =  2, 
X  +  72/+    2=1, 
3x  +  5y  -  5z  =  -3, 
4X-32/  +  22  =  1. 


8.  Obtain  a  consistent  system  of  equations  from  the  system  in  Ex.  7  by  replac- 
ing the  term  —3  by  a  new  value. 

9.  In  three  linear  homogeneous  equations  in  x,  y,  z,  w,  the  latter  are  proportional 
to  four  determinants  of  order  3  formed  from  the  coefficients. 

*  For  an  abbreviated  statement,  the  concepts  matrix  and  its  rank  are  needed.     Cf. 
Bocher,  Introduction  to  Higher  Algebra,  p.  46. 


CHAPTER  XII 
Resultants  and  Discriminants 

1.   Introduction.     If  the  two  equations 

ax  +  6  =  0,        cx-\-d^O  {a9^0,  c^O) 

are  simultaneous,  i.e.,  if  x  has  the  same  value  in  each,  then 

X  = = ,  R  =  ad  —  be  =  0, 

a  c 

and  conversely.  Hence  a  necessary  and  sufficient  condition  that  the 
equations  have  a  common  root  is  i?  =  0.  We  call  R  the  resultant  (or 
eliminani)  of  the  two  equations. 

The  result  of  eliminating  x  between  the  two  equations  might  equally 
well  have  been  written  in  the  form  6c  —  ad  =  0.  But  the  arbitrary 
selection  of  R  as  the  resultant,  rather  than  the  product  of  R  by  some 
constant  as  —1,  is  a  matter  of  more  importance  than  apparent  at  first 
sight.  We  seek  a  definite  function  of  the  coefficients  a,  6,  c,  d  of  the  func- 
tions ax  +  h,  ex  +  d,  and  not  merely  a  property  /?  =  0  or  i?  5^  0  of  the 
corresponding  equations.  Accordingly,  we  shall  lay  do^^^l  the  definition 
in  §  2,  which,  as  the  reader  may  verify,  leads  to  R  in  our  present  example. 

Methods  of  elimination  which  seem  plausil)le  often  yield  not  R  itself, 
but  the  product  of  R  by  an  extraneous  function  of  the  coefficients.  This 
point  (illustrated  in  Ex.  3,  p.  156)  indicates  that  the  subject  demands  a 
more  careful  treatment  than  is  often  given. 

We  may  even  introduce  an  extraneous  factor  zero.     Let  a  5^  0, 

/(x)  =  .T-  —  2  aX  —  3  a^,  g{x)  —  X  —  a. 

From  /  subtract  (x  +  a)g.  Multiply  the  remainder,  — 2a(x4-a),  by  x  —  3  a 
and  add  the  product  to  2  a/.  The  sum  is  zero.  But  the  resultant  is  —4:a^  (the 
value  of  /  for  X  =  a)  and  is  not  zero.  As  we  used  g  only  in  the  first  step  and  there, 
in  eflect,  replaced  it  by  x-  —  a^,  we  really  found  the  resultant  of  the  latter  and  /. 
The  extraneous  factor  introduced  {cf.  Ex.  7,  p.  152)  is  the  resultant  oi  x  -\-  a 
and  /  and  this  resultant  is  zero. 

150 


§21  RESULTANTS  AND  DISCRIMINANTS  151 

2.  Resultant  of  Two  Polynomials  in  x.     Let 

( /(a:)  -  aoa:"' +  aix'"-!  +  •  •  •  +  am  (ao  ?^  0) 

^  ^  ^  ^(a:)  =  box-  +  6ix"-i  +  •  •  •  +  6„  (6o  ^  0) 

be  two  polynomials  of  degrees  m  and  /i.  Let  cci,  .  .  .  ,  a™  be  the  roots  of 
f{x)  =  0.  Now  q:i  is  a  root  of  (j{x)  =  0  only  when  g{ai)  =  0.  The  two 
equations  have  a  root  in  common  if  and  only  if  the  product 

g{ai)g{a-z)   .   .   .  g{am) 

is  zero.  This  symmetric  function  of  the  roots  of  f(x)  =  0  is  of  degree  n 
in  any  one  root  and  hence  is  expressible  as  a  polynomial  of  degree  n  in  the 
elementary  symmetric  functions  (Chap.  VII,  §3),  which  equal  —ai/ao, 
02/00,  ....  To  be  rid  of  the  denominators  Oo,  it  suffices  to  multiply  our 
polynomial  by  ao".     We  therefore  define 

(2)  Rif,  g)=  c'o"g{ai)g{a2)  .  .  .  g{<xm) 

to  be  the  resultant  of  /  and  g.     It  equals  a  rational  integral  function  of 

Oo,     .     .     .    ,    dm,   Oo,     •     .     •    ,   On- 

EXERCISES 

1.  If  m  =  1,  n  =  2,  R{f,  g)  =  boUi^  —  biOoai  +  62^0^. 

2.  If  m  =  2,71=  1,  R{f,  g)  =  ao{boai  +  bi){boa-2  +  61)  =  aobi'^  -  o.Mi  +  OoW, 

since  Oo(ai  +  0:2)  =  —  Oi,  aoaia2  =  02. 

3.  If  /3i,  .  .  .  ,  /3„  are  the  roots  of  g{x)  =  0,  so  that 

fif(a.)=  6o(«i-/3i)(ai-/32)    .    .    .    («i  -  /3„), 

then 

R{f,  g)=  a^^'br  (ai  -  /3i)(a,  -  ^2)    •    •    •    ("i  -  /3n) 
•(«2-  ;8i)(«2-/32)    .    .    .    (a2-  fin) 


•  (am  —  /3l)(am  —  ^2)     ■    .    .    {otm  —  fin)- 

Multiplying  together  the  differences  in  each  column,  we  see  that 

/2(/,^)  =  (-l)'""&o"'M)/(^2)  .  .  .f{fin)  =  {-i)""'R{gJ)- 

4.  If  m  =  2,  n=  1,  R{gJ)=  bo'f{-bi/bo)=  a^bx^  -  aAby  +  a^bo^  which  equals 
Rif,  g)  by  Ex.  2.     This  illustrates  the  final  result  in  Ex.  3. 

5.  If    m  =  n  =  2,  R{f,  g)  =  OQ^o^arar  +  ao-bubiaia2{ai  +  a-^ 

+  ao'-^6o62(ai'  +  «2')  +  a^^'bi'aia.  +  ao'bMa,  +  a.)  +  ao^fti' 
=  6uW  —  bobiUiCh  +  bob2{ay^  —  2  0002)  +  61-0002  —  bAuoai  +  00-^2^ 
This  equals  R{g,f),  since  it  is  unaltered  when  the  o's  and  b's  are  interchanged. 


152  THEORY  OF  EQUATIONS  [Ce.  XII 

6.  R  is  homogeneous  and  of  degree  n  in  ao,  .  .  .  ,  Om',  homogeneous  uikI  of 
degree  in  in  bo,  ...  ,  bn.     R  has  the  terms 

ch"bn"',         {-iy""bo"'am". 

7.  R{J,g.g2)=R{!,g,)'R{f,g,). 

8.  7?(/,x")-(-l)""'a„.". 

3.  Irreducibility  of  the  Resultant  of  Two  Polynomials  in  One  Varia- 
ble.* The  resultant  of  two  polynomials /(x)  and  g{x)  was  seen  (§  2)  to 
equal  a  polynomial  r  (ao,  .  .  .  ,am,hQ,  .  .  .  ,  6„)  in  the  coefficients  of /and  ^. 
Let  these  coefficients  be  regarded  as  independent  variables.  Then  r  is 
irreducible,  i.e.,  is  not  equal  to  the  product  of  two  polynomials  ri  and  r^ 
in  Oo,  .  .  .  ,hn  with  numerical  coefficients,  if  neither  ri  nor  ri  is  a  numerical 
constant.**  Suppose  that  r  =  riVi.  Since  r  is  homogeneous  in  Oo,  .  .  .  ,  am, 
each  factor  ri  is.     Likewise,  each  r,  is  homogeneous  in  6o,  .  .  .  ,  6„.     Hence 


(''^'■••■^•'■(^■•••'d"'H''«o'--- j-'H'' 


Replace  ai/ao,  .  .  .  ,  am/ao  by  the  corresponding  symmetric  functions  of 
the  roots  ai,  .  .  .  ,  «„,  also  61/60,  •  •  •  ,  6„/6o  by  the  corresponding  sym- 
metric functions  of  /3i,  .  .  .  ,  /3„.  Let  the  factors  on  the  right  become 
the  polynomials  Pi  and  P2  in  ai,  .  .  .  ,  /3„.     Then  (Ex.  3), 

(ai  -  /3:)    .   .   .    («i  -  /3„)(a2  -  i^i)    .   .   .    (a„,  -  ^„)  ^  PjPo, 

identically  in  the  a's  and  /3's.  Apart  from  numerical  factors.  Pi  is  there- 
fore the  product  of  certain  of  the  differences  ai  —  /3i,  .  .  .  ,  and  P-i  the 
product  of  the  others.     But  this  is  impossible  since  Pi  is  sjTnmetric  in 

ai,  .  .  .  ,  am  and  symmetric  in  ^Si,  .  .  .  ,  /3„. 

4.  A  Correct  Conclusion  to  be  Drawn  from  Any  Method  of  Elimina- 
tion. Since  the  determination  of  r  by  means  of  symmetric  functions  of 
the  roots  is  excessively  laborious  unless  m  or  n  is  very  small,  we  shall  later 
give  other  methods.  But  we  shall  not  know,  without  a  careful  enquiry, 
whether  or  not  such  a  new  method  introduces  an  extraneous  factor.     Each 

*  In  place  of  §§3,  4,  the  reader  may  use  §  9.  But  this  substitution  should  be  made 
only  if  the  briefest  course  is  desired. 

**  This  is  evident  for  the  resultant  ad  —  6c  in  §  1.  For,  if  it  were  the  product  of  two 
Uuear  functions,  the  one  not  involving  a  would  necessarily  be  d  (or  a  numerical  constant 
times  d)  and  similarly  the  other  factor  would  then  be  a. 


§4J  RESULTANTS  AND  DISCRIMINANTS  153 

method  leads  in  fact  to  a  polynomial  F{ao,  .  .  .  ,  6„)  with  the  property 
that  every  set  of  solutions  ao,  .  .  .  ,  6„  of  r  =  0  is  a  set  of  solutions  of 
F  =0.     It  then  follows  that  r  is  a  factor  of  F. 

For  example,  if  R{f,  g)  —  0, 

/  =  a^~  +  ai.c  +  02  =  0,        g  =  b^x-  +  bix  +  62  =  0 
have  a  common  root  x.    Then 

hf  —  a^g  =  {ai)2  —  a2&o).'^-  +  (ai&2  —  a2h))x  =  0, 
— &o/+  oog  =  {aobi  —  aibo)x  +  00^2  —  a-ibo  =  0. 
Exclude  for  the  moment  the  case  a^  =  bo  =  0.    Then  x  9^  0  and 

0062     —02^0  CH^2     —  02^1 

Oo^i   —  ai6c         O0&2  —a-ibo 


(3)  F^ 


=  0. 


It  is  easily  verified  that  F  =  0  also  in  the  excluded  case.  Hence  any  set  of  solu- 
tions ao,  .  .  .  ,  62  of  r  =  0  is  a  set  of  solutions  of  F  =  0.  We  found  r  in  Ex.  5 
aloove.     It  is  seen  to  be  identical  with  this  F. 

To  prove  in  general  that  r  is  a  factor  of  F,  set 

r  =  cotto"  +  Ciao"-^  +  •  •  •   +  c„, 

where  Co,  .  .  .  ,  Cn  are  polynomials  in  ai,  .  .  .  ,  hn,  while  Co  is  not  identi- 
cally zero  (Ex.  6  above) .  Express  also  i^  as  a  polynomial  in  Go  and  apply 
the  greatest  common  divisor  process  to  F  and  r.  Suppose  that  r  is  not 
a  factor  of  F.     If  *  the  degree  of  F  in  ao  is  =  n,  we  may  write 

koF  =  qor  +  n,         hr  =  qin  +  r2,         hr^  =  qoV^  +  n, 

where  qo,  qi,  q^,  n,  r^  may  involve  ao,  while  ko,  ki,  ko,  rz  do  not  (for  sim- 
plicity we  assume  that  r^  is  the  first  r,  not  involving  ao).  If  r^  were  iden- 
tically zero,  r2  (or  a  factor  actually  involving  ao)  would  be  a  factor  of  r, 
as  shown  by  the  last  two  of  our  three  equations.  Since  ro  is  of  lower  degree 
in  ao  than  r,  this  contradicts  the  irreducibility  of  r  (§3).  Hence  there 
exist  constants  a/,  .  .  .  ,  bn  such  that 

r3(ai',  .  .  .  ,  hn')  ^  0,         Co  (a/,  .  .  .  ,  6/)  5^  0. 

For  tti  =  a/,  .  .  .  ,  hn  =  hn,  r  becomes  a  polynomial  in  ao  with  constant 
coefficients  and  hence   (Ch.  V)   vanishes  for  some  value  ao'  of  ao.     By 

*  In  the  contrary  case,  we  drop  the  first  equation  and  set  ri  =  F. 


154 


THEORY  OF  EQUATIONS 


[Ch.  XII 


hypothesis,  any  set  of  solutions,  as  ao'/u',  .  .  .  ,  6„'  of  r  =  0  is  a  set  of 
solutions  of  F  =  0.  Hence  F{ao,  .  .  .  ,  6„')  =  0.  For  these  values 
Co,  .  .  .  ,  bn  oi  Go,  .  .  .  ,  hn,  we  have  ri  =  0  by  the  first  of  our  three 
equations,  then  r-i  =  0  by  the  second,  and  n  =  0  by  the  third.  The  last 
result  contradicts  rsiai,  .  .  .  ,  bn)  9^  0. 

//  any  method  of  eliminating  x  between  two  equations  in  x  leads  to  a  rela- 
tion F  =  0,  where  F  is  a  polynomial  in  the  coefficients,  then  F  has  as  a  factor 
the  true  resultant  of  the  equations. 


6.   Sylvester's  Dialytic  Method  of  Elimination.     Let  the  equations 
aox^  +  aix-  -{-  aox  +  03  =  0,        hx"-  +  biX  +  60  =  0 

have  a  common  root  x,  so  that  their  resultant  r  is  zero. 

Multiply  the  first  equation  by  x  and  the  second  by  x-  and  x  in  turn. 
We  now  have  five  equations 

OoX'*  +  OiX^  +  02^""  +  asx  =  0, 

OqX^  +  ttix-  +  aox  +  03  =  0, 
box'  +  bix^  +  b.x''  =  0, 

bo3i^  +  61.T-  +  b2X  =  0, 

box-  +  b,x  +  62  =  0, 

which  are  linear  and  homogeneous  in  x\  x\  x~,  x,  1.     Hence 

OO       Ol       02       ^3       0 
0         Oo       Ol       ^2       fls 

(4)  F=     bo     Ih     b.     0      0 

0      bo     bi     Ih     0 
0      0      bo     bi     bo 

is  zero.     By  §  4,  r  is  a  factor  of  F.     But  the  diagonal  term  OoW  of  F  is  a 
term  of  r  (Ex.  G,  p.  152).     Hence  F  is  the  resultant. 
In  general,  if  the  equations  are 

aoX"'  +  •  •  •  -i-  am  =  0,     box""  +  •  •  •  +  6„  =  0, 
we  multiply  the  first  equation  by  rc"-\  a:"-^  .  .  .  ,  rr,  1 ,  in  turn,  and  the  sec- 


ond by  a:'"~S  x" 


,  X,  1,  in  turn.     We  obtain  n  +  m  equations  which 


§  51  RESULTANTS  AND  DISCRIMINANTS 

are  linear  and  homogeneous  in  the  w  +  n  quantities  x'"+"~^,  . 
Hence  the  determinant 


(5) 


Oo       Ctl       (I2    '    •    •    dm      0  .    .    .    . 

0      Oo     ai     a2  .  .  .  dm    0    , 
0       0       ao      «!      02  .    •    .   cim    0 


0 0      Oo    Oi     0-2    . 

bo     bi bn   0  .  . 

0      bo     bi b„ 


0 


0      bo     bi 


.  0 
.  0 


bn 


n  rows 


VI  rows 


155 

,  X,  1. 


is  zero.     By  §  4,  r  is  a  factor  of  F.    But  the  diagonal  term  Oo^bn!^  is  a 
term  of  r.     Hence  F  is  the  resultant. 


EXERCISES 

1.  For  m  =  n  =  2,  the  resultant  is 


Oo 

ai 

a2 

0 

0 

ao 

ai 

02 

ho 

61 

62 

0 

0 

60 

fci 

h. 

Interchange  the  second  and  third  rows,  apply  Laplace's  development,  and  prove 
that 

r  =  (0062)^  —  {aohi){aiho^, 

where  (ao&o)  denotes  00^2  —  «2&o,  etc.     Compare  with  (3). 

2.   For  m  =  n  =  3,  show  by  interchanges  of  rows  that 

Oo  tti  a2  03  0  0 

bo  61  b2  h  0  0 

0  Oo  ffi  02  as  0 

0  60  61  62  63  0 

0  0  Oo  Oi  02  03 

0  0  60  &i  bi  hi 


r  =  — 


156 


THEORY  OF  EQUATIONS 


[Ch.  XII 


Apply  Laplace's  development,  selecting  minors  from  the  first  two  rows,  and  to  the 
complementary  minors  apply  a  similar  development.  This  may  be  done  by 
inspection  and  the  following  value  of  —  r  be  obtained: 

(0061)1(0162)  (0263)  -  (0163)^+  (O263)(ao63)| 
—  (0062)  \  (0062)  {(hh)  —  (0063)  (0163)  \ 
+  (0063)  ]  (aobi)  (0263)  —  (0063)-  i . 

The  third  term  of  the  first  line  and  the  first  term  of  the  last  line  are  alike.  Hence, 
changing  the  signs, 

r  =  {aobsY  —  2  (ao6i)  (flo^s)  (0263)  —  (0062)  (('063)  (0163) 

+  (0062)-  (0263)  +  (rto6i)(ai63)"  —  (ao6i)  (0162)  (0263) . 

3.   For  m  =  n  =  3,  the  method  which  led  to  (3)  gives 

-  bof  +  aog  =  (ao6i)  x-  +  (0062)  x  +  (0063), 
(63/  —  a3g)/x  =  (0063)  X-  +  (0163)  X  +  (0263). 

By  (3),  the  resultant  of  these  two  quadratic  functions  is 


(0063)     (0061) 
(0263)     (0063) 


(0063)     (ao6i) 
(0163)     (0062) 


(0163)     (0062) 
(0263)     (0063) 


This  is,  however,  not  the  resultant  r  of  the  cubic  functions  /,  g.  To  show  that 
(0063)  is  an  extraneous  factor,  note  that  the  terms  of  F  not  having  this  factor 
explicitly  are 

{aj)i)  (0263)  1  (ao6i)  (0263)  —  (0062)  (0163)  I . 


The  quantity  in  brackets  equals  —(0063)  (0162),  since 

=  (ao6i)(a263)  -  (ao62)(ai63)  +  (ao63)(ai62). 


0  = 


Co  Ol  02  fl3 

60  61  62  63 

Oo  Ol  O2  fl3 

60  61  62  63 


We  now  see  that  F  =  r  •  (0063),  where  r  is  given  in  Ex.  2. 

4.  Verify  that  (0063)  is  an  extraneous  factor  by  showing  that  if  x^  —  1  =  0, 
x^  -  X  =  0,  then  r  =  0,  (0063)  ^  0. 

5.  The  resultant  oi  L  =  ax  +  ffy  and  L'  =  a'x  +  H'y  is  R  =  a^'  -  a'0.     The 
determinant  of  the  coefficients  of  x^,  xy,  y'  in  L-  =  0,  LL'  =  0,  L"-  =  0  is 


R'  = 


act  a^'  +  a' 13       /8/3' 

a"-  2  a'/3'  15'^ 


§  6)  RESULTANTS  AND  DISCRIMINANTS  157 

If  72  =  0  there  exist  values  not  both  zero  of  x  and  y  such  that  L  =  L'  =  0  and 
hence  values  of  x-,  xy,  y"^,  not  all  zero,  such  that  L'  =  0,  etc.  Thus  R  =  0  implies 
R'  =  0.  Since  R  is  irreducible,  it  is  a  factor  of  R'.  But  if  R'  =  0,  we  are  not  to 
infer  hastily  that  the  values  of  x"^,  xy,  y"^  obtained  from  the  three  equations  linear  in 
them  are  consistent  {i.e.,  the  product  of  the  first  and  third  equals  the  square  of 
xy)  and  hence  have  no  right  to  conclude  that  R'  =  0  implies  R  =  0  and  thus  that 
R'  is  a  power  of  R  (as  done  in  some  texts). 

If  R'  =  0,  the  three  linear  homogeneous  equations  whose  coefficients  are  the 
elements  in  the  three  rows  of  the  determinant  R'  have  solutions  not  all  zero,  which 
may  be  designated  x-,  xy  —  z,  if.     Then  the  equations  may  be  written  in  the  form 

U  =  1  CL&Z,      LL'  =  {a^'  +  a'0)z,      U-  =  2  a'^'z. 

Thus 

0  =  {LL'y-  -  UU'-  =  Rh\ 

li  R  9^  0,  then  2  =  0,  L  =  L'  =  0,  Rx  =  Ry  =  0,  whereas  x,  y,  z  are  not  all  zero. 
Hence  i?'  =  0  implies  /2  =  0.  Thus  each  irreducible  factor  of  R'  is  a  numerical 
multiple  of  R.     By  examining  one  term  of  R' ,  we  see  that  R'  =  R^. 

6.  The  determinant  of  the  coefficients  of  x^,  xhj,  xy"^,  y^  in 

L3  =  0,        L'L'  =  0,        LU^  =  0,        L'3  =  0, 

equals  R^.     Prove  as  in  Ex.  5  and  also  as  in  Ex.  7. 

7.  Reduce  the  determinant  R'  in  Ex.  5  to  the  form  R^.  If  /3  =  0,  R'  is  evidently 
R^.  li  0  ^  0,  multiply  the  elements  of  the  second  column  by  —a/0,  those  of  the 
third  column  by  a^//3'',  and  add  the  products  to  the  elements  of  the  first  column. 
The  elements  of  the  new  first  column  are  0,  0,  R^/0^.     Hence 


,_R^ 

2  a/3           /3 

_  ^' 

P 

al3'  +  a'0     0' 

~    P 

al3       0 


R'-R. 


8.  If  f or  F  =  0  we  omit  one  of  the  equations  in  §  5,  we  have  a  consistent  set 
of  equations  which  determine  x  in  general.  Thus  if  m  =  n  =  2,  .vf  =  0,  /  =  0, 
<7  =  0  give  ao{aobi)x  =  —  00(00^2).  The  latter  is  in  agreement  with  the  Unear 
equation  in  the  example,  p.  153. 

6.   Discriminants.     Let  ai,  .  .  .  ,  am  be  the  roots  of 
(6)  fix)  -  aox""  +  aix-"-!  +  ■  -  •  +  a^  =  0  (ao  7^  0) 

As  in  Ch.  Ill,  §  3,  we  define  the  discriminant  of  (6)  to  be 
D  =  ao^^-^Cai  -  a2)"(ai  -  a^)'  ...(«!-  aJ2(^^  _  ^3)2  ^   ^   _  (^^_^  _  ^j2. 

Evidently  D  is  unaltered  by  the  interchange  of  any  two  roots.  Since  the 
degree  in  any  root  is  2  (w  —  1),  the  symmetric  function  D  equals  a  poly- 
nomial in  Go,  ...  ,  ttm-     Indeed,  ao-'"~^  is  the  lowest  power  of  ao  sufficient 


158 


THEORY  OF  EQUATIONS 


ICh.  XII 


to  cancel  the  denominators  introduced  by  replacing  I,ai  by  —  ai/ao,  •  .  .  , 
aiaz  .  .  .  am  by  db  am/ao.     Now  * 

f\(Xi)=  ao(ai  —  a2)(ai  —  as)  .  .  .  («!  —  am), 
f'{a-i)  =  ao(o:2  —  ai){az  —  a^)  .  .  .  (0:2  —  "m), 
J'{a^=  ao(a3  —  «i)(a3  —  0:2)  (0:3  —  a^   .   .  .   (0:3  —  an^,  .... 

Hence 

ao'"-r(ai)    .   .   ./'(aJ=ao='"-U-l)^+-+-'-+'"-Kai-«2)^.   •   •  («m-i-«m)' 

=  (-1)       2        a^£). 

By  (2),  the  left  member  is  the  resultant  of /(.t),  J'{x).     Hence 

m(m— 1)    1 


(7) 


D=(-l) 


ao 


i^  (/,/'). 


For  another  proof  that  D  is  a  numerical  multiple  of  R/aQ,  see  Ex.  9  below. 


EXERCISES 

1.  Show  that  the  discriminant  of  /  s  y^  +  py  +  g  =  0  is  —  4  p'  —  27  q^  by 
evaluating  the  determinant  of  order  five  for  R{f,f'). 

2.  Find  the  relation  between  the  discriminant  of  f(x)  =  0  and  the  resultant  of 
7nf(x)  —  xJ'{x)  and/'(.r). 

3.  Hence  the  discriminant  of  ciqx^  +  Oix^  +  a-ix  +  03  is  —  |  r,  where 

?■  =  (oiOo  —  9  Ooas)-  —  (2  ({2-  —  6  aiQs) (2  ar  —  G  ooOa)' 

is  the  resultant  of  Uxx"^  +  2  a-ix  +  3  03  =  0,  3  oo-r-  +  2  a^x  +  02  =  0,  by  (3) . 

4.  The  discriminant  of  the  product  of  two  functions  equals  the  product  of  their 
discriminants  multiplied  by  the  scjuare  of  their  resultant.  Hint:  use  the  expres- 
sions in  terms  of  the  differences  of  the  roots. 

5.  Ofl  is  a  factor  of  R{f,  /')  by  the  first  column  of  its  determinant. 

6.  For  Oo  =  1,  the  discriminant  equals 


1      ai 
1      at 


ao"    .    .    .    012 


1 


^m       ^^rn 


Si  S2 

Si         S3 


.    .    Sm-l 
.    .    Sm 


"m— 1    S,n       Sm^l     •    •    •     Sim— 2 


where  Sj  =  ai'  +  •  •  •  +  a,„'.    See  Exs.  10,  11,  p.  141. 

*  By  differentiating  /(x)  =  ao{x  —  ai)  .  .  .  {x  —  am)  or  by  the  first  part  of  §  5, 
Ch.  VII. 


5  71  RESULTANTS  AND  DISCRIMINANTS  159 

7.  Hence  the  discriminant  oi  x^  -\-  px  -{-  q  =  0  equals 
3         0       -2p 
0      -2p    -3q     =-4:p^-27q\ 

-2p   -3q        2p2' 

8.  The  discriminant  D{ao,  .  .  .  ,  am)  is  irreducible.  As  in  §  3,  a  factor  would 
equal  a  product  P  of  powers  of  the  differences  oti  —  aj  such  that  P  is  symmetric 
in  ai,  .  .  .  ,  am.  Thus  every  difference  would  be  a  factor.  But  the  product 
of  the  first  powers  of  all  the  differences  is  changed  in  sign  by  any  interchange  of 
two  roots  (Ch.  XI,  §  12).     Hence  P  is  divisible  by  the  square  of  the  last  product. 

9.  Prove  that  Z)  is  a  constant  times  R{f,  /')  ^  ao  by  use  of  §  4.  Since  Z)  =  0 
impUes  R  =  0,  the  irreducible  D  is  a  factor  of  R.  But  D  is  of  total  degree  2  m  —  2 
in  oo,  Oi,  .  .  .  ,  and  R  is  of  total  degree  2  m  —  1.  Hence  R/D  is  of  the  first  degree 
and  thus  (Ex.  5)  a  numerical  multiple  of  oq. 

T.f  Euler's  Method  of  Elimination.  Let  /  and  g  be  given  by  (1). 
If /(x)  =  0  and  g{x)=  0  have  a  common  root  c,  then 

fix)  =  {x-c)  fi{x),        g{x)  =  (a;  -  c)  gi(x), 

identically  in  x,  where  fi{x)  is  a  polynomial  of  degree  m  —  1,  and  gi{x)  is 
of  degree  n  —  1.     Hence 

fix)gi{x)=g{x)fi{x), 

identically  in  x.  Hence  if  ao,  .  .  .  ,  6„  are  any  numbers  for  which  R(f,  g) 
=  0,  there  exist  constants  qi,  .  .  .  ,  q^,  pi,  .  .  .  ,  Pm  not  all  zero  for  which 

(aox^ -}- aix"'-'^ -{-  •  •  •   +  ajiqix"-^  -\-  qox""-^-  -\-  ■  •  •   +  q,,) 
^  {box^  +  6ix"-i  +  •  •  •  +  bn){pix'--'  +  ^2^;'"--+  -  -  -  -\-Pm), 

identically  in  x.     Equating  the  coefficients  of  like  powers  of  x  in  the  two 

products,  we  obtain  the  relations 

aoQi  -  hopi  =  0, 

fli(/i  +  aoq2  —  bipi  —  boP2  =  0, 

arnQn-l  +  am-iqn  —  &nPm-l  "  bn-lPm    =  0, 

OmQu  —    bnPm.  =   0. 

Since  these  m+n  linear  homogeneous  equations  in  the  unknowns  qi,  .  .  .  , 
Qn,  —Ph-  •  •  }  —Pm  have  a  set  of  solutions  not  all  zero,  the  determinant 
of  the  coefficients  is  zero.  Interchanging  the  rows  and  columns  of  this 
determinant,  we  get  (5).  The  proof  that  (5)  is  the  resultant  follows  as  in 
the  last  two  lines  of  §  5. 


160 


THEORY  OF  EQUATIONS 


[Ch.  XII 


8.  t   Bezout's  Method  of  Elimination.     When  the  two  equations  are 
of  the  same  degree,  the  metliod  will  be  clear  from  the  example 

/  =  aoX^  +  aix^  +  ttox  -\-  as  =  0,         g  =  boX^  -\-  bia:^  +  hx  +  63  =  0. 

Then 

dog  —  hof, 
(8)  (aox  +  ai)g  -  (b^x  +  61)  /, 

{aox-  +  aix  +  a2)g-(box'^  +  bix  +  62)/ 
equal  respectively 

{aobi)x-  +  (0062)  a:  +  (0063)  =  0, 

(0062)^;'-+  ]  (0063)  +  (0162)  \x-\-{aJ}s)  =  0, 
{aob3)x^  -\-{aib3)x  +(0263)  =  0, 

where  (ao&i)  =  Gobi  —  aibo,  etc.  The  determinant  of  the  coefficients  is  the 
negative  of  the  resultant  R{f,  g).  Indeed,  it  is  divisible  by  R  (§  4)  and 
has  a  term  of  —R.  The  negative  of  the  determinant  is  seen  to  have  the 
expansion  given  as  r  in  Ex.  2,  p.  156. 

The  three  equations  used  above  are  e\'ident  combinations  of 

x-f=0,     xf  =  0,    /  =  0,     .r-^  =  0,     xg  =  0,     g  =  0, 

the  latter  being  the  equations  used  in  Sylvester's  method  of  elimination.  The 
determinant  of  the  coefficients  in  these  six  equations  is 


R  = 


The  operations  carried  out  to  obtain  the  above  three  quadratic  equations  are 
seen  to  be  step  for  step  the  following  operations  on  determinants.  First,  a^t^R 
is  derived  from  the  determinant  R  by  nuiltij>lying  the  elements  of  the  last  three 
rows  by  a^.  To  the  eleniouts  of  the  new  fourth  row  add  the  products  of  the  ele- 
ments of  the  1st,  2nd,  3rtl,  5tli,  6th  rows  by  —60,  —b\,  —62,  «i,  «2  respectively 
Icorresponding  to  the  formation  of  the  third  function  (S)].  To  the  elements  of 
the  fifth  row  add  the  products  of  the  elements  of  the  2nd,  3rd,  6th  rows  by  —60,  bi, 
ai  respectively  Icorresponding  to  the  second  function  (8)1.  Finally,  to  the  ele- 
ments of  the  sixth  row  add  the  products  of  the  elements  of  the  third  row  by  —bo 
[corresponding  to  ai^  —  bof].     llcnce 


ao 

«! 

Oo 

03 

0 

0 

0 

ao 

fll 

02 

as 

0 

0 

0 

oo 

«1 

fl2 

fls 

bo 

b, 

^2 

63 

0 

0 

0 

bo 

^1 

b2 

63 

0 

0 

0 

bo 

61 

b2 

h 

RESULTANTS  AND  DISCRIMINANTS 


161 


Oo 

Ol 

02 

as 

0 

0 

0 

Oo 

Ol 

02 

as 

0 

0 

0 

Oo 

ai 

aj 

as 

0 

0 

0 

(0063) 

(0163) 

(0263) 

0 

0 

0 

(0062) 

(0063)  +  (0162) 

(0163) 

0 

0 

0 

(ao6i) 

(Oo?>2) 

{(lobs) 

ao'  R  = 


so  that  R  equals  the  3-rowed  minor  enclosed  by  the  dots.  The  method  of  Bezout 
therefore  suggests  a  definite  process  for  the  reduction  of  Sylvester's  determinant 
of  order  P.  n  (when  m  =  n)  to  one  of  order  n. 

Next,  for  equations  of  different  degrees,  consider  the  example 

/  =  Qqx'^  +  ttix^  +  aox-  +  Qzx  +  04,        g  =  60^^  +  6ix  +  62. 
Then 

aox^g  —  bof,         (aox  +  ai)x^g  —  (&oX  +  &i)/ 

equal  respectively 

(oofoi)  x^  +  (00^2)  X"  —  ttsbox  —  aA, 

(oofeo)  x^  +  I  (flifeo)  —  Qs&ol  a;2  —  '0361  +  0.460 1  X  —  0461. 

The  determinant  of  the  coefficients  of  x^,  x^,  x,  1  in  these  and  xg,  g,  after 
the  first  and  second  rows  are  interchanged,  is  the  determinant  of  order  4 
enclosed  by  dots  in  the  second  determinant  below.     It  is  the  resultant 

R{L  g)  by  §  4. 

As  in  the  former  example,  we  shall  indicate  the  corresponding  operations  on 
Sylvester's  determinant 

Oo      ai     02     as     04     0 


R 


u 

Oo 

ai 

fl2 

03 

04 

60 

61 

b-i 

0 

0 

0 

0 

bo 

b. 

62 

0 

0 

0 

0 

bo 

&1 

&2 

0 

0 

0 

0 

bo 

&1 

&2 

Multiply  the  elements  of  the  third  and  fourth  rows  by  oo.  In  the  resulting  deter- 
minant Ofl^i?,  add  to  the  elements  of  the  third  row  the  products  of  the  elements  of  the 
first,  second  and  fourth  rows  by  —60,  —  &i,  Oi  respectively.  Add  to  the  elements 
of  the  fourth  row  the  products  of  those  of  the  second  by  —bo.     We  get 


162 


THEORY  OF  EQUATIONS 


ICh.  XII 


Oo 

Oi 

02 

as 

a* 

0 

0 

ao 

Oi 

ao 

as 

Ga 

0 

0 

(0062) 

(0162)  -  0360 

—  0301  —  (libo 

—  0461 

0 

0 

(oobi) 

(00^2) 

—Uzbo 

—  0460 

0 

0 

bo 

61 

62 

0 

0 

0 

0 

60 

61 

62 

Of?  R  = 


Hence  i2  equals  the  minor  enclosed  by  dots. 


EXERCISES  t 

1.  For  m  =  S,  n  =  2,  apply  to  Sylvester's  determinant  R  exactly  the  same 
operations  as  used  in  the  last  case  in  §  8  and  obtain 

(0062)      (0162)  —  0360       —0361 

R  =     (ao6i)  (0062)  —  a36o 

60  bi  62 

2.  Hence  show  that  the  discriminant  of  Oax^  +  Uix"^  +  a^x  +  03  =  0  is 

2  0002     0102  +  3  O0O3     2  01O3 
Oi  2  02  3  03 

3  Oo  2  Oi  02 
=  18  a<)Oi0203  —  4  Oo02^  —  4  0/03  +  01^02-  —  27  00^03-. 

For  in  =  n  =  4,  reduce  Sylvester's  R  (as  in  the  first  case  in  §  S)  to 

(0061)  (0062)  (ao&s)  (ao&4) 

(0062)  (flo^s)  +  {(hb-i)  (0064)  +  (0163)  {(iibi) 
(ao&a)  («o64)  +  (0163)  (0164)  +  (0263)  (0264) 
(0064)  {aibi)  (0264)  (0364) 

4.   For  /  and  g  of  degree  n,  the  ith  function  (8),  when  ^VTitten  as  a  determinant 
of  the  second  order,  is  seen  to  equal 

diix"-'^  +  di2X"-- +  •  •  •  -\-din, 


where 
Then 


dij  =  (oo6,+;-i)  +  (oi6,+y_2)  + 


n(>i-l) 

2     D,    D 


R-i-l) 

On  1    •    .    •    djiii 

This  D  is  called  the  Bezout  determinant  of  /  and  g.    Show  that  dji  =  da. 


+  (oi_i6,). 
.  dm 


9)  RESULTANTS  AND  DISCRIMINANTS 

5.  Hence  verify  for  m  =  n  =  5  that  R  can  be  derived  from 


163 


(ao6i) 

(00^2) 

(«o63) 

(0064) 

(aoh) 

(0062) 

(00^3) 

{(lobi) 

(ctah) 

(aA) 

(0063) 

(0064) 

ioJ>b) 

(aA) 

(aA) 

(0064) 

(0065) 

(aA) 

(aA) 

(aA) 

(0065) 

(aih) 

(a-A) 

(a-A) 

(ciA) 

by  adding  to  its  nine  central  elements  the  elements  of 

(aifeo)  (0163)  (0164) 

(aA)  (aA)  +  (aA)     (aA) 

(aA)  (aA)  {asbi) 

6.  If  R(f,  g)  =  0,  we  obtain  a  consistent  set  of  equations  by  omitting  one  of 
Bezout's  equations.  Hence  they  determine  x.  If  m  =  /i  =  2,  find  x.  If  7n  =  n 
=  3,  find  X. 

7.  If  m  =  71,  set  gi{x)  =  x'"~"g{x).     Then 

Rif,  9)  =  R(f,  9i)  -^  (-l)'"("'-")a^— ». 

8.  If  m  =  n,  R{cf  +  dg,  sf  +  tg)  =  ±{ci  -  dsY'R{f,  g).      [Find  the  new  {aih,).] 

9.  Express  as  a  determinant  of  order  m  the  resultant  of /(x)  =  0  and  x"*  =  1. 
[Multiply/  by  x  and  reduce  by  x"'  =  1;  repeat.] 

9.t  Without  employing  the  results  of  §§3,  4,  we  may  give  a  direct 
proof  that  the  determinant  (5)  is  the  resultant  of/  and  g,  given  by  (1). 
While  the  method  is  general,  we  shall  present  it  only  in  the  case  m  =  3, 
n  =  2.     In  the  equation 


(9) 


tto 

Oi 

^2 

a^  —  z 

0 

0 

Oo 

Ql 

tti 

03- 

z 

bo 

h. 

hi 

0 

0 

0 

ho 

hy 

62 

0 

0 

0 

&0 

61 

62 

0, 


take  z  =  f{0i) .  Multiply  the  elements  of  the  first  four  columns  by 
jSi"*,  /3/,  /Si^,  j8i,  respectively,  and  add  the  products  to  the  last  column. 
All  of  the  elements  of  the  new  last  column  are  zero.  Hence  /(/3i)  and 
/(jSa)  are  the  roots  of  (9).     Since  the  equation  is  of  the  form 

60V  +  (    )2  +  F  =  0, 


164  THEORY  OF  EQUATIONS  ICn.  xii 

where  F  is  given  by  (4),  we  have 

Hence  the  Sylvester  determinant  F  is  the  resultant  R(J,  g). 
Moreover,  the  equation  in  z  is  the  eliminant  of 

g{x)   =Q,       Z  =f{x), 

and  hence  gives  explicitly  the  equation  obtained  from  g(x)  =0  by  apply- 
ing the  transformation  z  =  f{x)  of  Tschirnhausen  (Ch.  VII,  §  13). 

10.  t  Theorem.  Necessary  and  sufident  conditions  that  f(x)  and  g(x) 
shall  have  a  common  divisor  of  degree  d,  hut  none  of  higher  degree,  are  R  =  0, 
Ri  =  0,  .  .  .  ,  Rd-i  =  0,  Rd  9^  0,  where  R  is  the  determinant  (5),  and  Rk  is 
the  determinant  derived  from  R  by  deleting  the  last  k  rows  of  a's,  the  last  k 
rows  of  b\s,  and  the  last  2  k  columns. 

For  example,  if  7n  =  n  =  4, 

Oo      Oi      Qo      03      04      0 


(10) 


Ri  = 


0 

Go 

Ol 

02 

az 

04 

0 

0 

Oo 

fli 

02 

03 

bo 

61 

h 

^3 

b, 

0 

0 

bo 

bi 

62 

63 

64 

0 

0 

bo 

bi 

&2 

63 

To  prove  the  theorem  for  the  case  d  =  1,  set 

/i  =  pix'"--  +  •  •  •  +  pm-h    gi  =  gix"-2  -1- 

The  conditions  for  an  identity  of  the  form 

(11)  fgi-gfi^cx  +  c' 

are 

flog.  —  &0P1 


+  (7n-l 


aiqi  +  00(72 


bipi  —  60P2 


=  0, 
=  0, 


amqn-2  +  Om-lQ'n-l 


—   hnPm-2   —   hn-\Pm-\    =   C, 
—    bnPm-1        =   C'. 


§  10]  RESULTANTS  AND  DISCRIMINANTS  165 

Omitting  the  last  equation,  we  liave  m  -\-  n  —  2  linear  equations  for  the 
same  number  of  unknowns  qi,  —pi-  The  determinant  of  the  coefficients 
equals  Ri  with  the  rows  and  columns  interchanged.  Hence  if  i^i  ?^  0 
we  may  choose  c  =  Ri  and  find  values  not  all  zero  of  the  unknowns  satis- 
fying all  of  the  above  equations  except  the  last,  and  then  choose  c'  so  that 
the  last  holds.  Let  R  =  0.  Then  /  and  g  have  a  common  linear  factor, 
but  no  common  factor  of  degree  >  1  since  the  right  member  of  (11)  is  of 
degree  unity. 

But  li  R  =  Ri  =  0,  we  may  take  c  =  0  and  find  values  not  all  zero  of 
Qi,  Pi  satisfying  all  but  the  last  of  the  above  equations.  The  resulting 
value  of  c'  is  zero  by  (11),  with  c  =  0,  since/  and  g  have  a  common  factor 
X  —  r.    Then 

^  -.or  -  ~^U  - 0. 


X  —  r  X  —  r 

Since  not  all  of  the  m  —  1  linear  factors  of  the  first  fraction  are  factors 
of  /i  (of  degree  m  —  2),  at  least  one  is  a  factor  of  the  second  fraction. 
Hence  \i  R  =  Rx  —  (i,  J  and  g  have  a  common  factor  of  degree  >  1. 

To  prove  the  theorem  for  d  =  2,  we  employ  functions  J^.  and  g2  of  de- 
grees m  —  3  and  n  —  3,  respectively.     Of  the  conditions  for  the  identity 

(12)  Jgo  -  gfo  =  ca:2  +  ^'x  +  c", 

we  omit  the  two  in  which  c'  and  c"  occur  and  see  that  the  determinant 
of  the  coefficients  of  the  remaining  equations  is  Ro.     Then  if 

R  ^  Ri  =  0,      R.  ^  0, 

we  may  take  c  =  Ro  and  satisfy  all  of  the  conditions  for  (12).  Thus 
/  and  g  have  no  common  factor  of  degree  >  2. 


EXERCISES  t 

1.   By  performing  on  (10)  exactly  the  same  operations  as  used  in  §  8  to  reduce 
a  determinant  of  order  6  to  one  of  order  3,  show  that 

(0063)  (0064)  +  {aih)         {aihi)  +  (0063) 

Ri  =      (00^2)  (00^3)  +  {axbij         {(lobi)  +  (a  163) 

(aobi)  (ao^o)  (0063) 

Note  that  if  04  =  64  =  0,  the  present  work  reduces  to  the  former. 


166 


THEORY  OF  EQUATIONS 


[Ch.  XII 


2.   In  the  notation  of  Ex.  4,  p.  162,  the  preceding  Ri  with  its  first  and  third  rows 
interchanged  becomes  Di: 


Ry  =    -A. 


dn 

(/l2 

dn 

c/21 

c/22 

d-iz 

C^31 

dsz 

dzz 

3.   For7/i  =  n, 


T>k 


dn 


di  „-k 


d,,-. 


dn-k  n-k 


4.  Hence,  if  m  =  n,  f  and  g  have  a  common  divisor  of  degree  d,  but  none  of 
degree  >  d,  if  and  only  if />  =  0,  A  =0,  ...  ,  Dd-i  =  0,  Dj  9^  0. 

5.  Give  a  direct  proof  of  Ex.  4  by  multiplying  the  ith  function  in  Ex.  4,  p.  162, 
by  a  variable  ?/,-  and  summing  for  i  =  1,  .  .  .  ,  ^     Thus 

g-laoUi+iaox  -}-  ai)y2+  •  •  •  +(aox'-i+  •  •  •  )ytl  -  f'\b(iiji+{bo.t  +  hi)y2+  •  •  •] 

=  5ix"-i  +  52X"--+   •  •  •   +5n, 

where  5i  =  dnVi  +  •  •  •  +  d^yt,  .  .  .  ,  5„=  dmyi  +  •  •  •  +  dmyi. 

The  determinant  of  the  coefficients  of  7/1,  ...  ,  yt  in  5i,  .  .  .  ,  St  is  Dn~t.  If 
Z)  =  0,  take  i  =  n;  then  we  can  choose  ?/i,  .  .  .  ,  i/„  not  all  zero  so  that  5i  =  0, ,  .  .  , 
dn  =  0.  Then  gfi  —  fgi  =0  for  functions  /i  and  gi  of  degree  w  —  1,  so  that  / 
and  g  have  a  linear  divisor.  If  also  Di  =  0,  take  t  =  71  —  1;  then  we  can  make 
5i  =  0,  .  .  .  ,  5n-i  =  0.  Hence  gfo  —  fg2  =  5n  for  functions  Ji  and  ^2  of  degree 
n  —  2.  Since/  and  g  have  a  common  divisor,  the  constant  5n  is  zero,  and  hence 
they  have  a  common  divisor  of  degree  ^  2.     But  if  Di  7^  0,  we  can  make 

gh  -  f92    =   5«-lX  +  5„,       Sn-l9^   0, 

so  that  the  only  common  divisor  is  linear. 


MISCELLANEOUS  EXERCISES  167 

MISCELLANEOUS  EXERCISES 

1.  Find  a  necessary  and  sufficient  condition  that  the  roots  a,  /3,  7  of 
x^  +  px'^  +  gx  +  r  =  0  shall  be  in  geometrical  progression. 

2.  For  the  same  equation  find  2q:^/3^.     [Repjace  x  by  1/x.] 

3.  Find  the  equation  with  the  roots  a^  +  /3-,  a^  +  7^,  /S^  +  7^. 

4.  Find  the  equation  with  the  roots  a~  +  /3'^  —  7^,  a-  +  7"  —  /3",  etc. 

5.  Find  the  equation  with  the  roots  a-  +  «/3  +  0',  etc. 

6.  Solve  the  equation  in  Ex.  1  by  forming  and  solving  the  quadratic  equation 
with  the  roots  (a  +  co/3  +  0)^7)^  and  {a  +  co'/3  +  wyY,  where  co-  +  w  +  1  =  0. 
(Lagrange.) 

7.  Solve  x^  —  28  X  +  48  =  0,  given  that  two  roots  differ  by  2. 

8.  Find  a  necessary  and  sufficient  condition  that 

/(x)  =  x^  -\-  px^  +  5.C'  +  r.c  +  s  =  0 

shall  have  one  root  the  negative  of  another  root.     When  this  condition  is  satisfied, 
what  are  the  quadratic  factors  of /(x)? 

9.  Solve  J{x)  =  x^  —  Q  x^  -\-  13  .r^  —  14  .r  +  6  =  0,  given  that  two  roots  a 
and  j8  are  such  that  2  a  +  /3  =  5.  Hint:  f(x)  and  /(o  —  2  x)  have  a  common 
factor. 

10.  Diminish  the  roots  oi  x'*  -{-  qx-  -\-  rx  -\-  s  =  0  (s  ^  0)  by  such  a  number 
that  the  roots  of  the  transformed  equation  shall  be  of  the  form  a,  in /a,  h,  m/h,  and 
show  how  the  latter  equation  may  be  solved. 

11.  Solve  x"  -  2  x2  -  16  .r  +  1  =  0  by  the  method  of  Ex.  10. 

12.  By  use  of  the  equation  whose  roots  are  the]  squares  of  the  roots  of 
x^  +  a;'  —  x2  +  2a;  —  3  =  0  and  Descartes'  rule,  show  that  the  latter  equation 
has  four  imaginary  roots. 

13.  Similarly,  x^  +  .r^  +  8  .c  +  6  =  0  has  imaginary  roots. 

14.  If  all  of  the  roots  of  x"  +  ax"~^+  6x"~^  +  •  •  •   =0  are  real, 

a2-26>0,     62_2ac  +  2d>0,     c^  -  2  6d  +  2ae  -  2/>  0,  .  .  .  . 

Hint:  Form  the  equation  in  ?/  =  x^. 

15.  Solve  x^  +  px  +  5  =  0  by  eliminating  x  between  it  and  x-  -\-  vx  -{■  w  =  y 
by  the  greatest  common  divisor  process,  and  choosing  v  and  w  so  that  in  the  result- 
ing cubic  equation  for  y  the  coefficients  of  y  and  y-  are  zero.  The  next  to  the  last 
step  of  the  elimination  gives  x  as  a  rational  function  of  y.  (Tschirnhausen,  Acta 
Erudit.,  Lipsiae,  H,  1683,  p.  204.) 

16.  Find  the  preceding  y-cnhio,  as  follows.  Multiply  x"^  +  vx  -\-  w  =  y  hy  x 
and  replace  x^  hy  —px  —  q;  then  multiply  the  resulting  quadratic  equation  in  x 
by  X  and  replace  x^  by  its  value.  The  determinant  of  the  coefficients  of  x^,  x,  1 
must  vanish. 

17.  Eliminate  y  between  y^  =  v,  x  =  ry  -\-  sy"^,  and  get 

x'  —  3  rsvx  —  (r^v  +  s^v~)  =  0. 

Take  s  =  1  and  choose  r  and  v  so  that  this  equation  shall  be  identical  with  x'  +  px 
-\-  q  =  0,  and  hence  solve  the  latter.     (Euler,  1764.) 


168  THEORY  OF  EQUATIONS 

18.  Eliminate  y  between  1/  =  v,  x  =  f  -{-  ey  +  y-  and  get 

1  e        f  —  X 


e       f  —  X        V 
J  —  X        V  ev 


=  0. 


This  cubic  equation  in  x  may  be  identified  with  the  general  cubic  equation  by  choice 
of  e,  /,  V.     Hence  solve  the  latter. 

19.  Determine  r,  s  and  v  so  that  the  resultant  of 

7/3  =  V, 

shall  be  identical  with  x^  +  px  -\-  q  =  0.     (Bezout,  1762.) 

20.  Show  that  the  reduction  of  a  cubic  equation  in  x  to  the  form  y^  =  y  by  the 
substitution 

_  r-\-  sy 

""'  1  +  2/ 

is  not  essentially  different  from  the  method  of  Ex.  18.  [Multiply  the  numerator 
and  denominator  of  .c  by  1  —  ?/  +  ?/-.] 

21.  If  the  discriminant  of  a  cubic  equation  is  positive,  the  number  of  positive 
roots  equals  the  number  of  variations  of  signs  of  the  coefficients. 

22.  Descartes'  rule  gives  the  exact  nmiiber  of  positive  roots  only  when  all  the 
coefl&cients  are  of  Uke  sign  or  when 

each  Pi  being  =  0.  Without  using  that  rule,  show  that  the  latter  equation  has  one 
and  only  one  positive  root  r.  Hints:  There  is  a  positive  root  r  by  Ch.  I,  §  12 
(a  =  0,  6  =  00 ).  Call  F{x)  the  quotient  of  the  sum  of  the  positive  terms  by  x', 
and  call  —  A7'(x)  that  of  the  negative  terms.  Then  A' (x)  is  a  sum  of  powers  of 
1/x  with  positive  coefficients. 

If        x>r,        P{x)>P{r),        .V(.c)  <  .Y(r),        /(.r)  >  0; 

If        X  <  r,        P(x)  <  F(/-),        .V(x)  >  .V(r),        /(x)  <  0.     (Lagrange.) 

23.  If /(x)=  /i(x)  +  •  •  •  +/a(.c),  where  each  /j(.c)  is  like  the /in  Ex.  22,  and 
if  ii  is  the  greatest  of  the  single  positive  roots  of  /i  =  0,  .  .  .  ,  fk  =  0,  then  R  is 
an  upper  Umit  to  the  positive  roots  of  /  =  0. 

24.  Any  cubic  or  quartic  equation  in  x  can  be  transformed  into  a  reciprocal 
equation  by  a  substitution  x  =  ry  +  s. 

25.  Admitting  that  an  equation  f(x)  =  x"-\-  •  •  •  =0  with  real  coefficients 
has  n  roots,  show  algebraically  that  there  is  a  real  root  between  a  and  b  if 
/(a)  and /(6)  have  opposite  signs.  Note  that  a  pair  of  conjugate  imaginary  roots 
c  ±  di  are  the  roots  of  (x  —  c)^  +  (/'-  =  0  and  that  this  quadratic  function  ia 


MISCELLANEOUS  EXERCISES 


169 


positive  if  x  is  real.  Hence  if  Xi,  .  .  .  ,  Xr  are  the  real  roots  and  </)(x)  =  (x  —  Xi) 
.  .  .  {x  —  Xr),  then  4>{a)  and  4>{b)  have  opposite  signs.  Thus  a  —  Xi  and  b  —  Xi 
have  opposite  signs  for  at  least  one  real  root  x,.     (Lagrange.) 

26.   If  s,  is  the  sum  of  the  jth  powers  of  the  roots  of  an  equation  of  degree  n 
and  if  m  is  any  integer,  the  equation  is 

1 


X" 


X" 


Sm+n+1        Sm-{-n 


.    X 

.    Sm+2        Sm+1 


Sm+2  n—1       S„j-|-2  71—2 


Sm+n      Sm+n—l 


=  0. 


Hint:  Use  the  second  set  of  Newton's  identities.     (Jacobi.) 
27.   li  a  <  b  <  c  .  .  .    <  I,  and  a,  0,  .  .  .  ,  \  are  positive, 

X 


X  —  a 


+ 


X  —  b 


+ 


re  —  c 


+ 


+ 


X  —  I 


-\-t  =  0 


has  a  real  root  between  a  and  &,  one  between  b  and  c,  .  .  .  ,    one  between  k  and  i, 
and  if  t  is  negative  one  greater  than  I,  but  if  t  is  positive  one  less  than  a. 

28.  Verify  that  the  equation  in  Ex.  27  has  no  imaginary  root  by  substituting 
r  +  si  and  r  —  si  in  turn  for  x,  and  subtracting  the  results. 

29.  In  the  problem  of  three  astronomical  bodies  occurs  the  equation 

r'+iS-  m)/-"  +  (3  -  2  ,.y  -  ixr''  -  2  ^.r  -  ^  =  0, 

where  0  <  m  <  1.    Why  is  there  a  single  positive  real  root?    As  ju  approaches 
zero,  two  complex  roots  and  the  real  root  approach  zero. 

30.  Discuss  the  equation  obtained  from  the  preceding  by  changing  the  signs  of 
the  coefficients  of  r^  and  r. 


31.  By  Newton's  identities. 

1 

0 

Pl 

S3  =    - 

Pi 

1 

2p, 

=  -Pi'  +  Sp 

P2 

Pi 

3p3 

1 

0 

0 

...  0     Pi 

Pi 

1 

0 

...  0     2p2 

Sk=  - 

P2 

Pi 

1 

...  0     3p3 

Ps 

P2 

Pi 

...  0    4p4 

Pk- 

1    Pk- 

-2    Pk- 

i  .  .  .  Pi   kpk 

where  all  but  the  last  term  in  the  main  diagonal  is  1,  and  all  terms  above  the  diagonal 
are  zero  except  those  in  the  last  column.     If  A;  >  n,  we  must  set  py  =  0{j  >  n). 


170 


THEORY  OF  EQUATIONS 


32.   By  Newton's  identities, 

1     0    si 
3!p3  =  -|  si    2    S2 

Si      Si      S3 


Jc\p,=  - 


1 

0 

0   . 

.  0 

Si 

Si 

2 

0   . 

.  0 

S2 

S2 

Si 

3   . 

.  0 

S3 

Sk- 

1  Sa-2 

Sk-3    . 

.  .  k 

Sk 

iik  =  n.    But  if  A;  =  7i, 

Sk 


Sk-l 
Sk+l       Sk 


Sk-2 
Sk-l 


Sk-n 
Sk-n+l 


=  0. 


Sk+n      Sk+n-1      Sk+n-2       .    .    .    Sk 

33.  Let  Si  =  ai^  +  •  •  •  +  aj.    Let  ar,  .  .  .  ,  «„-  be  the  roots  of 

yn  ^  p^yn-1  +    .    .    .    +  p^^  =  0. 

Set  y  =  a:/  and  multiply  the  result  by  ay^'^- " ,  where  k  =  2  7i.    Sum  f  or  j  =  1 ,  . 
Thus 

Hence 


,n, 


Sk  +  PlSk-2  +  P^Sk-i  + 


+  PnSk-2n  =  0. 


Sk  Sk-2 

Sk+l       Sk-l 


Sk-i 
Sk-3 


Sk-2  n 
Sk-2n+l 


0. 


Sk+n       Sk+n-2      Sk+n-i      •    •    ■    Sk-n 

34.   Obtain  a  vanisliing  determinant  similar  to  that  in  Ex.  33  but  having  the 
subscripts  of  the  s's  in  each  row  decreased  by  3. 

10     0     0 

0 


35. 


Si 

Si 

S2 

S3 

= 

S3 

Si 

So  Si  S2 
So  Si  S2  S3 
Si   S2   S3  Si 

1       Vi  P2  Ps 

0  So  Si  +  Pi  So  S2  +  PiSi  +  poSo 

So        Si  +  PlSo      S2  +  PiSi  +  P2S0      S3  +  P1S2  +  V-iSi  +  P3S0 
Si        S2  +  PiSi      S3  +  P1S2  +  P2S1      Si  +  P1S3  +  P2S2  +  P3S1 

1  Vi  Pi  V3 

On  (71  -  l)ih      (n  -  2)]h 

n      (n  —  l)7;i      (n  —  2)p2      (n  -  3)p3 
Pi     2p2  3p3  4  2^4 


MISCELLANEOUS  EXERCISES  171 

36.  If  n  =  3,  the  last  determinant  may  be  obtained  from  the  Sylvester  resultant 
R  oi  x^  -\-  pi.r^  +  Ihx  +  ]h  and  its  derivative  by  multiplying  the  elements  of  the 
first  row  of  jR  by  —3  and  adding  tlie  products  to  the  elements  of  the  third  row. 

37.  Express  the  determinant  of  order  4  in  the  Sj  (analogous  to  the  first  one  in 
Ex.  35)  as  a  determinant  of  order  6  in  the  p's.  For  n  =  4,  identify  the  latter  with 
the  resultant  of  x^  +  jhx^  +  P-^^~  +  Ih^^  +  Pi  and  its  derivative. 

38.  Let  Sk  be  the  sum  of  the  kih  powers  of  the  roots  .Ci,  .  .  .  ,  a;„  of  a  given 
equation.  The  coeflScients  of  the  equation  having  as  its  roots  the  5  n(n  —  1) 
squares  of  the  differences  of  the  x's  can  be  found  from  Si,  S2,  .  .  .  ,  where  Sp  is 
the  sum  of  the  pth  powers  of  the  roots  of  the  latter  equation.  Expand  by  the 
binomial  theorem 

(X  -  Xi)2P  +  (X  -  XoyP  +    .    .    .    +  (x  -  XnY^, 

set  X  =  Xi,  .  .  .  ,  X  =  Xn  in  turn,  add  and  divide  by  2.     Thus 

(2p)(2p-l) 
Op  =  ns2p  —  2  ps2p-iSi  -\ ~-^ S2P-2S2 

,2p(2p-l)  ■  .  .  (p  +  1)     , 
-  "  '   ^~-  1-2  ...  p  '^"• 

(Lagrange.) 

39.  In  particular,  *Si  =  ns2  —  sr,  S2  =  WS4  —  4  S1S3  +  3  S2^ 

S3  =  nse  —  6  S1S5  +  15  S2S4  —  10  S3''.  Hence  give  the  equation  whose  roots  are 
the  squares  of  the  differences  of  the  roots  of  a  given  cubic  equation.  Deduce  the 
discriminant  of  the  latter. 

40.  The  equation  whose  roots  are  the  n(?i  —  1)  differences  Xj  —  Xk  of  the  roots 
of  /(x)  =  0  may  be  obtained  hy  eliminating  x  between  the  latter  and  f{x  -\-  y)  =  Q 
and  deleting  the  factor  y"  (arising  from  y  =  xj  —  xj  =  0)  from  the  eliminant. 
The  equation  free  of  this  factor  may  be  obtained  by  eliminating  x  between  /(x)  =  0 
and 

l/(x  +  2/)-/(.r)|/y=/'(x)+r(.r)^+  •  •  •  +/"(.r)  ^  ^^^""'  _  ^^  =  0. 

This  eliminant  involves  only  even  powers  of  y,  so  that  if  we  set  y"^  =  z  \nq  obtain 
an  equation  in  z  having  as  its  roots  the  squares  of  the  differences  of  the  roots  of 
/(x)  =  0.     (Lagrange.) 

4L   Compute  by  Ex.  40  the  ^-equation  when/(.i-)  =  x^  +  px  +  g. 

42.   Except  for  h  —  Q,  the  equation 


a  —  X        b 
b        f-x 


0, 


has  a  real  root  exceeding  a  and  /,  and  one  less  than  a  and  /.     [Substitute  a  and  / 
for  X  in  turn]. 


172 


THEORY  OF  EQUATIONS 


43.   Let  the  equation  in  Ex.  42  have  distinct  real  roots  a,  /3,  where  a  >  fi.     Then 
there  are  three  real  roots  of* 

a  —  X        b  c 


D{x)  - 


f-x 

g 


9 
h  —  X 


=  0. 


Hint:  The  results  of  substituting  a  and  /3  for  x  in  D(x)  are 

[c  V^^f  +  g  V^^]',         -  [c  Vf-i5-  g  Va  -  0]\ 

where  the  product  of  the  radicals  in  each  is  +6.  Hence  if  neither  a  nor  /3  is  a 
root,  there  is  a  root  >  a,  one  <  /3,  and  one  between  a  and  /3.  If  a  is  a  root, 
there  is  a  root  <  /3  and  hence  thi-ee  real  roots. 

44.   li  a  =  fi  in  Ex.  43,  then  a  =  /  is  a  root  of  D{x)  =  0  and  there  are  two 
further  real  roots. 


45. 


aa'  +  bb'  +  cc' 
ae'  +  bj'  +  eg' 
b'  a' 


-\-be 


r  c 


ea'+jb'  +  gc' 
ee'+Jf'  +  gg' 
b'  c' 

r  g' 


+  bg 


+  ce 


=  af 

c'  a' 
g'  e' 


a'b' 
e'f 


+  ag 


+  cf 


c'  b' 

g'  r 


Combine  the  first  and  third,  second  and  fifth,  fourth  and  sixth: 


a  b 

a'  b' 

+ 

a  c 

a'  c' 

+ 

b  c 

^ 

b'  c' 

e  J 

e'  r 

e  g 

e'  g' 

J  g 

r  g' 

a  b 

2         a  c 

2          b  c 

= 

+ 

+     , 

e  J 

e  g 

f  g 

46.  Hence,  in  particular, 
a^  +  62  -(-  c2        ae  +  bf  -{-  eg 
ae  +  bf-\-  eg        e-  +  p  -f  g- 

47.  Hence  if  a,  b,  c  and  e,  f,  g  are  the  direction  cosines  of  two  lines  in  space,  and 
if  e  is  the  angle  between  them,  so  that  cos  d  —  ae  -\-  bf  -\-  eg,  then  sin'-  d  equals  the 
above  sum  of  three  squares. 

48.  For  the  determinant  in  Ex.  43, 

o2  +  ^2  +  c2  —  .1-2      ab  -\'  bf  +  eg  ae  -\-  bg  +  eh 

D{x)  'D{-x)=     ab  +  bf  +  eg  b'-  +  P  +  (72  -  .1-2    be  +  Jg  +  gh 

ac-\-bg  +  eh  be  -\-  fg  +  gh  c-  +  ^2  _|_  /,2  _  ^2 

=  -.T«  +  x\a'  +P  +  h'  +  2b-  +  2e-  +  2 g-)  -  x'-iD,  +  D,  +  Ds)  +  D2(0), 

*  This  theorem  is  important  in  many  branches  of  pure  and  applied  mathematics. 
Besides  this  proof  and  that  in  Ex.  48,  other  more  advanced  proof.s,  including  that  by 
Borchardt,  are  given  in  Salmon's  Modern  Higher  Algebra,  pp.  48-56. 


MISCELLANEOUS  EXERCISES  173 

where  D3  is  the  first  determinant  in  Ex.  46  for  e  =  b  and  Di  and  D2  are  analogous 
minors  of  elements  in  the  main  diagonal  of  the  present  determinant  of  order  3 
with  X  =  0.  Hence  the  coefficient  of  —x-  is  a  sum  of  squares  (Ex.  49).  Since 
the  function  of  degree  6  is  not  zero  for  a  negative  value  of  x^,  D{x)  =  0  has  no 
purely  imaginary  root.  If  it  had  an  imaginary  root  r  +  si,  then  D{x  -\-  r)  =  0 
would  have  a  purely  imaginary  root  si.  But  D{x  +  r)  is  of  the  form  in  Ex.  43  with 
a,  f,  h  replaced  by  a  —  r,  f  —  r,  h  —  r.  Hence  D{x)  =  0  has  only  real  roots. 
The  method  is  applicable  to  such  determinants  of  order  n.     (Sylvester.) 

49.  In  Ex.  48,  Di -\-  Do  +  D3  equals 

(af  -  br-  +  {ah  -  c-y-  +  (fh  -  g-'-y  +  2  {ag  -  bcY  +  2  (of  -  bgY  +  2  {bh  -  cg)\ 

50.  Without  using  its  solution  by  radicals,  prove  that 

.f*  +  bx^  +  cx"  +  dx  +  e 

has  a  factor  x-  —  s.r  +  p,  where  s  is  a  root  of  a  sextic  equation,  and  that  p  is  a 
rational  function  of  s  and  the  coefficients. 

Hints:  There  are  six  functions  like  s  =  Xi-{-  Xt,  next, 

C  =   SX1X2  =  s{Xz  +  .1-4)  +  J)  -\-  XzXi, 

—d=  Sx-iX2X3  =  SX3X4  +  (.I's  +  Xi)p. 

Replace  Xs  +  .r4  by  —b  —  s  and  solve  for  p  the  resulting  linear  equations  in 
X3X4  and  p.  The  case  b  -\-  2  s  =  0  may  be  avoided  by  starting  with  another  pair 
of  roots. 

51.  Prove  Ex.  50  by  di\dding  the  quartic  by  the  quadratic  function  and  requir- 
ing that  the  linear  remainder  shall  be  zero  identically, 

52.  Prove  Ex.  50  by  use  of  (3)  and  (8)  in  Ch.  IV. 

53.  x^  +  bx^  +  cx^  +  dx^  -{-  ex'^  -{-  fx  -^  g  has  a  factor  x^  —  sx  +  P,  where  s  is  a 
root  of  an  equation  of  degree  15,  and  p  is  a  rational  function  of  s  and  the  coefficients. 
Hints:    Write 

0-1  =  X3  +  X4  +  Xs  +  a.-6,   <T2  =   X3X4  +  •  •  •  ,   as  =   X3X4X5  +  •  •  •  ,   0-4  =  X3X4X5X6 

for  the  elementary  symmetric  functions  of  x^,  .  .  .  ,  Xe,  and  show  that 

—  b  =  S  +  ai,  C  =  p  +  Sffi  -\-  0-2,  —d  =  pai  +  S<T2  +  0-3, 

e  ^  2}(T2  +  So-3  +  Oi,  —f=    Sai  +  pa3,      g  =  pcTi. 

The  first  four  relations  determine  the  o-'s.  Then  the  last  two  give  a  cubic  and  a 
quadratic  equation  in  p,  by  means  of  which  we  may  express  p  as  a  rational  function 
of  s  and  then  obtain  an  equation  in  s  alone.     Why  must  this  be  of  degree  15? 

54.  If  Ex.  53  were  solved  as  in  Ex.  51  (if  the  quotient  of  x*' +  •  •  •  byx-  +  •  •  • 
be  denoted  by  x*  —  cnx^  +  crox"^  —  asx  +  0-4,  we  obtain  the  above  six  relations), 
why  could  we  conclude  that  any  equation  of  degree  six  with  real  coefficients 
has  two  complex  roots  (independently  of  the  fundamental  theorem  of  algebra)? 


174 
55. 


THEORY  OF  EQUATIONS 


«1  +  «2  +  «3 

ai"-f~  a2'-{-  0:3^ 


ai  +  «2  +  "3 

56.   The  determinant  in  Ex.  55  equals 


=  2:(ai-ao)2 
3 


x\' 


«2 
"1     «2" 


-?! 


1     a2 


+ 


1     ai 
0:2    cci^ 


=  2 


[See  Ex.  46. 


Ct\  +  "2 


3    I    ai  4"  a2   ai'^H"  «2 


0=2 


57.   For  n  roots,  7i  =  3, 


D 


n 

Sl 

S2 

Sl 

S2 

S3 

V 

S2 

S3 

S4 

=  ^      a,- 


i,j,  k  =  1,  .  . 
i,  j,  k  distinct. 


'  ^^Y 


z,=  X 


i<j<l: 


Add  the  six  determinants  given  by  the  permutations  of  fixed  i,  j,  k.     Then 

1+1     +1  "i  +  «;■  +  «/t       oci^  +  aj~  +  afc- 

ar+  a/+  ar      ai^  +  a/  +  a>t' 
«i'+  «/+  «A,-'      "i-*  +  «/  +  ak* 

1  "i  "i" 
1  aj  af 
1      ayt      "A;" 


=  X 


i<j<k 


«t  +  «/  +  «t 

01  21  o 

Oil  ~r  «;  +  «/t 
111' 

oti      «;      at    j 

9  2  2 

"i      ay      «r  ! 


2(Q;i  —  (xj)'{ai  —  a;t)^(aj  —  a^.)^. 


58.  Comparing  the  theorems  in  Exs.  55  and  57  and  their  extensions  wath  Ex.  12, 
p.  102,  we  see  the  nature  of  a  proof  of  Borchardt's  Theorem:  An  equation  of  degree 
n  with  real  coefficients  and  distinct  roots  has  as  many  pairs  of  imaginary  roots  as 
there  are  changes  in  signs  in  the  series 


So  =  n, 


So 

Sl 

Sl 

S2 

So 

Sl 

S2 

Sl 

S2 

S3 

S2 

S3 

Si 

Sn-1 
Sn 


Sn—l     Sn 


S2n-2 


If  two  consecutive  terms  are  zero,  the  theorem  may  fail,  as  x^  +  1  =  0  shows. 
But  it  holds  if  an  isolated  zoro  occurs  and  is  suppressed. 

59.  Denote  the  last  series  by  Di  =  so,  D2,  D3,  .  .  .  ,  Dn-  There  are  exactly 
r  di.stinct  roots  of  the  given  equation  of  degi*ee  n  if  and  only  if  Dr  is  the  last  non- 
vanishing  determinant  of  this  series.  For,  as  in  Exs.  55-57,  Dk  is  the  sum  of  the 
various  products  of  tlie  squares  of  the  difYerences  of  k  of  the  roots  a\,  .  .  .  ,  a„. 
U  k  >  r,  each  product  involves  two  equal  as  and  hence  Dk  =  0.  If  A;  =  r,  the 
only  term  not  zero  is  that  involving  the  r  distinct  as,  so  that  Dr  7^  0.  (L.  Baur, 
Malh.  Annalen,  vols.  50,  52.) 


MISCELLANEOUS  EXERCISES 


175 


60.  The  n  roots  are  all  real  and  distinct  if  and  only  if  D2,  .  . 
positive.     (Weber,  Algebra,  2d  ed.,  I,  p.  322.) 

61.  If  each  cv  is  real  and  if  the  numbers 


,  Z>„  are  all 


Co, 


Co 

Ci       . 

Cn 

Ci 

C2       . 

Cn+\ 

Cn 

Cn+l- 

C2n+2 

+ 

C2 

^2n  — 

0 

are  positive,  all  of  the  roots  of 

Co  +  CiX  +  CiX^  +    • 

are  imaginary,  and  all  but  one  of  the  roots  of 

Co  +  cix  +  CoX-  +  •  •  •  +  C2„+ia;2"+i  =  0 

are  imaginary.     (Van  Vleck,  Annals  of  Math.,  4  (1903),  p.  191.) 

C2 1      C2  t+i 

C2t+1    C2t+2 


62.   The  results  in  Ex.  61  follow  if  the  Cti  and 


are  all  positive. 


(Kellogg,  Annals  of  Math.,  9  (1907),  p.  97.) 

63.  If  the  terms  with  negative  coefficients  in  an  equation  of  degree  n  are  —  ax""", 
— /3x"~*,  —  7X"~<',  .  .  .  ,  no  positive  root  exceeds  the  sum  of  the  two  largest  of 
the  numbers 

v  a,         V /3,         V  7,  ...  .  (Lagrange.) 

64.  In  Ex.  63,  no  positive  root  exceeds  the  greatest  of  the  numbers 

y/ka,         Vkl3,  .  .  .  , 


where  k  is  the  number  of  the  negative  coefficients  —a, 


(Cauchy.) 


65.*  Define  W  as  in  Ch.  IX,  §  8,  and  let /(a)  ^  0,  f{b)  ^  0.  If /(x)  =  0  has 
imaginary  roots,  Va  —  Vb  cannot  give  the  exact  number  of  real  roots  in  every 
interval  [a,  b] ;  but,  if  f{x)  =  0  has  no  imaginary  roots,  Va  —  Vb  gives  the  exact 
number  of  real  roots  in  every  interval  [a,  b].     Hint:  Use  (14),  Ch.  IX. 

66.  Budan's  Theorem  gives  the  exact  number  of  real  roots  of  f(x)  =  0  in 
[a,  b]  if /(rt)  9^  0,/(6)  9^  0,  iirovided  that,  for  r  =  0,  1,  .  .  .  ,  n  —  2,  real  roots  of 
/('■^(x)  —  0  separate  those  of  f^'"^'^Kx)  —  0  in  that  interval  from  each  other  and  from 
a  and  6.  The  term  "separate"  here  excludes  the  case  of  coincidence.  Hint:  At 
a  root  oi  f^'"^^^(x)  =  0,  the  functions  f^^'Kx)  and  f^'"^^Kx)  must  be  of  opposite  sign. 

67.  Descartes'  Rule  gives  the  exact  number  of  real  roots  only  when  Budan's 
Rule  is  exact  for  every  positive  interval  [a,  b].  Thus  it  is  exact  for  an  equation 
having  only  real  roots. 

68.  We  define  as  generalized  Sturm's  functions  for  an  interval  [a,  b]  a  sequence 
of  polynomials /(x),/i(x),  .  .  .  , /^(x),  with  the  following  properties: 

*  The  author  is  indebted  to  Professor  D.  R.  Curtiss  for  Exs.  65-72. 


176  THEORY  OF  EQUATIONS 

(a)  No  two  consecutive  functions  vanish  simultaneously  at  any  point  of  [a,  b]; 
(6)  fr{x)  does  not  vanish  in  [a,  b]; 

(c)  When,  for  1  =  i  ^  r  —  1,  fi{x)  vanishes  for  a  value  of  Xi  in  [a,  b],  fi-i{xi)  and 
/t+i(a;i)  have  opposite  signs; 

(d)  When/(.r)  vanishes  for  a  value  .ri  in  [a,  b],  fi{xi)  has  the  same  sign  as/'(xi). 
Prove  that  the  number  of  real  roots  of /(.r)  =  0  in  [a,  b]  is  equal  to  the  difference 

between  the  numbers  of  variations  of  signs  in  such  a  sequence  for  x  —  a  and  for 
X  =  b. 

Prove  the  corresponding  statement  for  an  interval  [c,  d]  within  [a,  b]. 

69.  Prove  that  generalized  Sturm's  functions  for  any  interval  [a,  b],  where 
a  and  b  are  both  positive  or  both  negative  and  f{x)  =  0  has  no  multiple  roots,  may 
be  obtained  as  foHows:  Take/i(.c)  =/'(x).  Arrange /(.r)  and/i(a;)  in  ascending 
powers  of  x,  and  divide  the  former  by  the  latter  (using  negative  powers  of  x  in  the 
quotient,  if  necessary) ;  let  the  last  remainder  of  degree  equal  to  that  of  fix)  be 
designated  by  r^ix);  then  /2(.c)  =  —ro{x)  -r-  x'^.  Define /i(.r)  similarly  by  division 
of  fi-2(.x)  by  fi-i(x),  both  being  arranged  according  to  ascending  powers  of  x; 
the  last  remainder  of  degree  ecjual  to  that  of  /i-2(-r)  is  divided  by  —x-  and  the 
quotient  taken  as  fi{x).  Show  that  the  sequence  thus  obtained  is  valid  for 
[—GO,  go],  provided  no  one  of  the  functions  vanishes  for  x  =  0. 

70.  Prove  that  generalized  Stunn's  functions  for  any  interval  [a,  b],  where 
a  and  b  are  both  positive  or  both  negative  and  /(.r)  =  0  has  no  multiple  roots,  may  be 
obtained  by  the  greatest  common  di\asor  process  for/(.c)  =  oox" +OiX"~^+  •  •  •  +a„ 
and/i(x),  with  the  signs  of  the  remainders  changed  (as  in  Sturm's  method),  if  we 
take 

Mx)  =  4>(x)  s  a,x"-'  +  2aoa;"-2  +   .  •  •  +  jiUn  (x  <  0), 

but  Mx)  =  -<p{x)  if  X  >  0.         Hint:  .r/'(.r)  +  </.(x)  =  ?;/(.r). 

71.  Prove  the  analogue  of  Ex.  69  when/i(.r)  is  taken  as  in  Ex.  70. 

72.  For  the  cubic  /(x)  =  a^^  +  UiX^  +  a^x  +  as  without  multiple  roots,  discuss 
the  validity  of  the  sequences  in  Exs.  69-71  for  any  interval  [a,  b],  where  a  <  0, 
&  >  0.  Hint :  If  as  ^  0,  discuss  whether  variations  of  signs  for  x  very  near  zero 
and  negative  =  variations  of  signs  for  x  veiy  near  zero  and  positive. 


ANSWERS 


Page  2. 

1.  1.6,4.4.  2.  No  real.  4.   1.2,-1.8,-3.4. 

Page  7. 

2.  2.1.  3.  (-0.845,  4.921),  (-3.155,  11.079);  between  -4  and  -5. 
4.  1.1,  —1.3.  5.  Between  0  and  1,  0  and  —1,  2.5  and  3,  —2.5  and  —3. 
9.   120  {x^  +  x),  120  x^  -  42. 

Page  9. 
I.  3.  2.  2,  —2.  3.    —1.  4.   Double  roots  1,  3. 

Page  10. 

3.  Use  Ex.  3,  p.  9,  abscissas  —1,  3.  4.   Use  Ex.  2,  p.  9. 

Page  II. 
I.  One.         2.   Three.  3.   Three.  4.   1,  1,  -2.  5.  One. 

Page  16. 
7.  0.3,1.5,-1.8.        8.   1.2.        9.   1.3,1.7,-3.0.         15.   1,2,3,-6. 

Page  23. 

3.  TV(19^-9),     ^'~J+^?"^^     i(6  +  V5-3r  +  2V5t). 

4.  Commutative  and  associative  laws  of  addition  and  multiplication. 
Distributive  law. 

Page  24. 

1.  ±(3  +  4i).  2.   ±(5  +  6t).  3.   ±(3-2^■). 
4.   ±[c  +  d  +  (c  -  d)z].  5.   ±(c  -  di). 

Page  26  (middle). 

2.  —  3,  —  3w,  —  3a;^;  I,  coi,  co-i. 

3.  cos^  +  zsin  A     (A  =  40°,  160°,  280°). 

177 


178  THEORY  OF  EQUATIONS 

Page  26  (bottom). 

1.  -1,  cosA  +  isiiiil     (A  =  36°,  108°,  252°,  324°). 

Page  30. 

2.  5,  -1  ± V-3.  3.    1  ±  i,  1  ±V2. 

4.  a:^  -  7  a;2  +19  a:  -  13  =  0.  5.   r*  _^  (1  _^  i^^2  -|-  1  =  0. 

7.    ±1,2  ±  V3.         8,  9.    V3,  2  ±  /.         10.   a;3  -  I  x2  -  I  a;  +  I  -  0. 

Page  32. 

2.  -5,  H5±V^).  3-    -6,  ±V^.  4.    -2,  l±i. 

5.  i  f  (-2±V-3). 

Page  34. 
I.  Three.  2.  Two.  3.   Two. 

Page  35. 
I.    -4,  2zhV3;  3,  3,  -6.      2.   Page  37.      3.   1.3569,  1.6920,   -3.0489- 
Page  37. 

1.  Sce3,  p.  35.  2.    -1.2017,  1.3300,   -3.1284. 

3.  1.24698,   -1.80194,   -0.44504.       4.    1.1642,   -1.7728,   -3.3914. 

Page  39. 

2.  -1,   -2,2,  3.  3.    1,  -1,  4±  V6. 

Page  43. 

2.  l±v^2,   -1±V^.       3.   4,   -2,   -l±i.       4.   See  Ex.  3,  p.  39. 

Page  53. 

I.   z  =  —I  —  2i,  ooz,  o}^z. 

Page  56. 

I.  x'  -  8^2  +  16  =  0.  2.  1,  3.                3.    4,  1  -V^. 

5.  2+\/3,  x2  +  2x  +  2  =  0.  6.  1,2.       7.    -3,1,5.       8.    4,  f,  -|. 

9.  1,  3,  5.       10.   2,  -6,  18.  II.  5,  2,  -1,  -4.       12.    1,  1,  1,  3. 

13-  Ps  =  ViJh'  16.  if  —  V2y  -  12  =  0. 

Page  58. 

3.  6,  4.   2.  5.   3. 


ANSWERS  179 

Page  6i. 
I.    1,  3,  6.  2.   2,  -1,  -4,  5.  3.    -12,  -35.  4.   2,  2,  -  3. 

Page  62. 

1.  3,    1,   O,    J.  2.     X,    2>    3-  3'  6-  4'  4>  4>   2- 

Page  65. 

I.  g-  —  2  pr  +  2  s.  2.   p-g  —  2  g^  —  p^-  -)_  4  s. 

3.  p'^  —  4  p-g  +  2  g-  +  4  pr  —  4  s. 

4.  ?/^  -  (p^  -  2  g)2/2  -f  (52  _  2  pr)y  _  ^2  =  0. 

5.  2/3  —  qy"^  +  pr?/  —  r^  =  0.  6.   rif  -\- 2  qif  -\-  4:  py  +  8  =  0. 
7.  ^i2  -  2  ^2.                     8.   ^lE's  -  3  E3.  9.   ^1^2. 

10.  Ei^  -  S  E1E2 -\- 3  E3.  II.  Ei'-SEiEi. 

12.  ^1^3  -  4  ^4  if  n  >  3,  £Ji^3  if  n  =  3. 

Page  71. 

2.  SaSbScSd  —USaSbSc+d  +  2IlSaS6+c+d  +2Sa+6S,.+d  —  6  Sa+ft+c+rf,  if  «,   &,   C,   d  aVQ 

distinct;  but  if  all  are  equal, 

j\  {Sj  —  6  Sa%a  +  8  SaSsa  +  3  S2a^  —   6  S4  a). 

3.  See  Exs.  1,  2,  12,  13,  page  65. 

Page  76. 
3.   ?/  -7qi/-^  14  gV  _  7  qSy  =  c.  4.    e'"  -  2  e^'"  (?«  =  0,  .  .  .  ,  4). 

Page  77. 

..  P'-^P'l  +  5pr  +  l\  ,.2p'-2q.  8.    -p'  +  24r. 

r  —  pq  I        r  1  A'     I 

3  p'^if  —  4  p^r  —  4  g^  —  2  pqr  —  9  r^ 
^'  (r  -  pqf 

10.   27  r^  —  9  pgr  +  2  g^  =  0.       12.   y  =  q-\-  r/x.       13.   re  =  ,   ,    .f^- 

2  +  2y 

Page  83. 

I.   1,  -I  (1  ±\/^),  i  (7  ±V45).      2.   1,  a;2  +  i  (l  ± V5)a:  +1=0. 
3.    ±1,  a;2  ±  a:  +  1  =  0.  5.   2^  _^  2^  _  2  ^  _  1  =  0. 

6.  2^  +  2^  -  4  2^  -  3  22  _j_  3  2;  -1-  1  =  0.        8.   2  cos  2  7r/7,  etc. 


180  THEORY  OF  EQUATIONS 

Page  88. 


2. 


g  =  2,     r  +  r8  -I-  ri2  _|_  ^s^  g^^.,     2^-{-z^-4:Z-{-l=0. 


Page  98. 
I.   One,  between  -2  and  -3.  2.   One,  between  1  and  2. 

Page  99. 

I.   Ex.  2,  p.  37.       2.    (0,  1),  (-1.1,  -1).       3.  a;=  -i/inEx.  1,  p.37. 

4.    (0,  1),  (-2,  -1).        5.    (1,  2),  (-7,  -6).        6.    (0,  1),  (3,  4). 

Page  103. 
!•   2,  —2.  4.   1,  1,  two  imaginary. 

Page  105. 

I.    (-2,  -1),  (0,  1),  (1,  2).  2.    (-4,  -3),  (-2,  -1),  (1,  2). 

Page  113. 
I.  Page  117.  2,  3.   Exs.  1,  3,  page  119. 

Page  119. 

1.  -1.7728656.  2.  y  =-xm  Ex.  3,  p.  35. 
3.  Single,  -2.46954.        4,  5.  Exs.  2,  3,  p.  37. 

6.  Two  negative  and  2.121 +  ,  2.123 +  . 

7.  3.45592,  21.43067.       8.  2.24004099. 

Page  121. 

2.  Darwin's  quartic:    -12.4433  ±  19.7596  ^. 

3.  -0.59308,  -2.04727,  1.32048  ±  2.0039  ^•. 

4.  0.35098,  12.75644,  32.0602,  34.8322. 

Pages  123-24. 

1.  4.0644364,     -0.89196520,     0.82752156. 

2.  §1,     -1.04727  ±  1.13594  i.        5.     -2.46955. 

Page  128. 
I.  a;  =  5,     y  =  Q.  2.  a;  =  2,     y  =  1.  3.  x  =  a,    y  =  0. 


ANSWERS  181 

Page  130. 

1.  X  =  —8,     y  =  —7,     2  =  26.  2.  a;  =  3,     y  =  —5,     2  =  2. 

Page  134. 

2.  X  =  Q,     y  =  Z,    z  =  12.  3.  a:  =  5,     ?/  =  4,     2  =  3. 

^  (fe  -  b)(c  -  k)  ^  k(b-k)(c-k)(k-\-h  +  c) 

'^'  ^      {a-h){c-a)'  ^'  ^      a{h-a){c-a){a  +  h  +  c)' 

Page  137. 

1.  —a'>J)iCidi  +  a-ihicdi  +  a-ihzCicU  —  a-ih^ddi  —  a-zhiCids  +  a-ihiCzdi. 

2.  +,     +. 

Page  148. 

2.  Consistent:  2/  =  —8/7  —  2x,     2  =  5/7  (common  line). 

3.  Inconsistent,  case  (j8). 

4.  Inconsistent  (two  parallel  planes). 

5.  Consistent  (single  plane). 

Page  149. 

I.  a:  :  ?/  :  2  =  — 4  :  1  :  1.  2.  x  :  |/  :  2  =  — 10  :  8  :  7. 

3.  Two  unknowns  arbitrary.  4.  x  :  y  :  z  :  w  =  Qt  :  ?>  :  12  :  \. 

11  19  10         17 

S-z=-^x-—y,     w=-—x-  —  y. 

6.  y  =  -8/7  -2a;,     2  =  5/7. 

7.  Inconsistent  (determinant  4th  order  9^  0). 

Page  167. 

I.  ph  =  q^.  2.  3  r-  —  3  pqr  +  cf".     3.  Eliminate  a:  by  ?/  =  S2  —  x^. 

5.  Eliminate  a;  by  p-  —  g  +  px  =  ?/.        7.  2,  4,  —  6. 

8.  yqr  —  p'S  —  r-  =  0,     X"  -\-  rjp,     x^  •\-  px  ■\-  ps/r. 

9.  1,  3,  1  ±1.  12.  2^  +  224  +  5  23  +  3  22-22-9  =  0. 
13.  2^  +  15  2-  +  52  2  -  36  =  0.  89.  See  Ex.  17,  p.  78. 


INDEX 

(The  numbers  refer  to  pages.) 


Abscissa,  1 
Absolute  value,  24 
Amplitude,  24 
Argument,  24 
Axes,  1 

Bend  point,  3,  9,  11 
Bezout  on  elimination,  160 

—  determinant,  162 
Binomial  equation,  84 
Borchardt's  theorem,  174 
Budan's  theorem,  103,  175 

Cardan's  formula?,  32 

Cauchy:  symmetric  fimctions,  78 

Columns  of  determinant,  128 

ahke,  130,  138 

Common  divisor,  8,  95,  164 
Complementary  minors,  141 
Complex  number,  21,  47 

,  geometrical  representation,  24 

Conjugate  imaginary,  22,  29 
Continued  fraction,  125 
Continuity,  12,  51 
Coordinates,  1 
Cube  roots,  26,  35 

of  unity,  23,  24 

Cubic  equation,  10,  11,  16,  17,  31,  40,  75, 
80,  90,  99,  119,  122,  167-8 

Decagon,  88 

De  Moivre's  quintic,  76 

—  theorem,  25 

Derivative,  5,  69,  93,  96,  103,  106,  110, 

158 
Descartes  on  the  quartic,  42 

—  rule  of  signs,  105,  168,  175 
Determinant,  127,  134,  154-175 
Diagonal  term,  134 


Discriminant,  33,  41,  157 

—  of  cubic,  33,  78,  99,  158-9,  162 

—  of  quartic,  41,  45,  100 
Divisor,  58,  164 

Double  root,  8  (see  Discriminant) 
Duphcation  of  cube,  90 

Element  of  determinant,  128 

EUminant,  150 

Ehmination,  152-160 

Equal  roots,  8,  11,  33,  34,  45 

Equation  of  squared  differences,  78,  125, 

171 
Euler  on  elimination,  159 

quartic,  44 

Expansion  of  determinant,  130,  133,  138 

Factor  theorem,  8 

—  of  determinant,  131 
Ferrari  on  the  quartic,  38 
Fold,  8 

Foui'ier  on  roots,  106 

Fundamental  theorem  of  algebra,  47 

Geometrical  (see  Complex) 

—  construction,  16,  87,  90,  92 
Griiffe  on  finding  roots,  121 
Graph,  2,  14 

Graphical  solution,  2,  15-17,  110 

Homogeneous  equations,  148 
Horner's  method,  115 

Imaginary  number,  21 

—  roots,  28,  102,  107,  120,  124,  174-5 
Inflexion  point,  9,  44 

—  tangent,  10 
Integral  root,  59 

Intercihange  in  determinants,  132-4,  137-8 
Interpolation,  111 


183 


184 


THEORY  OF  EQUATIONS 


Interval,  97 
Invariant,  42 
Irrational,  30 
Irreduc'ible  case,  34,  35 
Irreducibility  of  resultants,  152 
Isolated,  95 

Lagrange  on  the  quartic,  40 
— ,  solution  of  equations,  125 
Laplace's  development,  141-3 
Linear  equations,  127-9,  144-9 
Lower  limit  to  roots,  58 

Minor,  129 

Modulus,  24,  25 

Moivre  (see  De) 

Multiple  root,  8  (see  Discriminant) 

Newton's  formula;,  70,  169,  170 

—  method  of  solution,  109-115 
for  integral  roots,  59 

Ordinate,  1 

Pentagon,  88 

Periods,  85 

Plotting,  2 

Polar  coordinates,  19 

Polynomial,  3,  6 

— ,  sign  of,  14 

Primitive  root,  28 

Product  of  determinants,  143 

Quadratic  equation,  15 

Quartic,  16,  38,  42,  80,  99,  120,  173 

Quintic,  76,  80,  83 

Radian,  27 

Rank  of  determinant,  145 

Rational  root,  61 

—  integral  function,  3 
Reciprocal  equation,  81 
Reduced  cubic,  10,  31 

—  quartic,  42 
Regula  falsi,  111 
Regular  polygon,  27,  87-90 

Relations  between  roots  and  coefficients, 
32,  39,  55,  Ch.  VII,  169,  170 


Relatively  prime,  28 

Remainder  theorem,  8     ■• 

Resolvent  cubic,  38,  39 

Resultant,  150 

RoUe's  theorem,  93 

Roots,  theorems  on,  13,  15,  30,  34,  39,  45, 

47,  55,  59,  61,  64,  69,  76,  90,  93,  96,  100, 

103,  105,  112,  174-6 

—  of  unity,  27 
— ,  nth,  26 

Rows  of  determinant,  128 
ahke,  138 

2-polynomial,  63 

2-function,  68,  70 

Slope,  3,  6 

Solution  of  numerical  equations,  109 

Solvable  by  radicals,  75,  84 

Square  root,  23 

Sturm's  functions,  98 

,  generalized,  176 

—  theorem,  95-102 
Sum  of  roots,  39,  56 

powers  of  roots,  69,  72,  169 

Sylvester  on  eUmination,  154 
Symmetric  function,  63,  121 

—  in  all  roots  but  one,  76 
Synthetic  division,  116 

Taylor's  theorem,  6,  113 
Transformed  equation,  10,  115,  119 
Trigonometric  form,  24 

—  solution  of  cubic,  34,  36 
Trinomial  equation,  11 
Triple  root,  8 
Trisection  of  angle,  90 
Tschirnhausen,  79,  164 

Upper  hmit  to  roots,  57,  175 

Variation  in  signs,  95 

Vector,  18 

Vieta  on  the  cubic,  31 

Waring's  formula,  72 


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